Phase Contrast Enhancement with Phase Plates in Electron Microscopy

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 138

Phase Contrast Enhancement with Phase Plates in Electron Microscopy KUNIAKI NAGAYAMA Okazaki Institute for Integrative Bioscience, National Institutes of Natural Sciences, 5‐1, Higashiyama, Myodaiji‐cho, Okazaki, 444‐8787, Japan

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . A. Phase Objects . . . . . . . . . . . . . . . . . . . . . . . B. A Trick to Convert Phases to Magnitudes . . . . . . . . . . . . . II. Issues in Phase Recovery in TEM . . . . . . . . . . . . . . . . . A. Historical . . . . . . . . . . . . . . . . . . . . . . . . B. Amount of Charges in the Phase Plate . . . . . . . . . . . . . . C. Remedies for Phase Plate Charging . . . . . . . . . . . . . . . D. Spatial Filters for Phase Contrast . . . . . . . . . . . . . . . . III. Dedicated Phase Plate TEM . . . . . . . . . . . . . . . . . . . A. Transfer Lens Doublet. . . . . . . . . . . . . . . . . . . . B. Phase Plate Holder . . . . . . . . . . . . . . . . . . . . . C. EVect of the Phase Plate Heating Holder . . . . . . . . . . . . . IV. Zernike Phase Contrast TEM . . . . . . . . . . . . . . . . . . A. Contrast Transfer . . . . . . . . . . . . . . . . . . . . . B. Optimum Material for Phase Plates . . . . . . . . . . . . . . . C. Comparison Between Zernike Phase Contrast and Defocus Phase Contrast . D. Biological Applications . . . . . . . . . . . . . . . . . . . 1. ZPC‐TEM Images of Virus Species . . . . . . . . . . . . . . V. Hilbert DiVerential Contrast TEM. . . . . . . . . . . . . . . . . A. Contrast Transfer . . . . . . . . . . . . . . . . . . . . . B. Comparison of Hilbert DiVerential Contrast and Defocus Phase Contrast . . C. Significance of Lower Frequency Components . . . . . . . . . . . D. Biological Applications . . . . . . . . . . . . . . . . . . . 1. Cultured Cells . . . . . . . . . . . . . . . . . . . . . 2. Cyanobacterial Cells . . . . . . . . . . . . . . . . . . . 3. Isolated Organelles . . . . . . . . . . . . . . . . . . . . VI. Foucault DiVerential Contrast TEM . . . . . . . . . . . . . . . . A. Contrast Transfer . . . . . . . . . . . . . . . . . . . . . 1. Foucault Knife‐Edge Scanning Filters . . . . . . . . . . . . . 2. Theory of Biased Derivative Filters . . . . . . . . . . . . . . 3. Knife‐Edge Scanning Filters that Realize Biased Derivative Filters . . . B. Numerical Simulations . . . . . . . . . . . . . . . . . . . VII. Complex Observation in TEM . . . . . . . . . . . . . . . . . . A. Basic Scheme and CTF Demodulation . . . . . . . . . . . . . . B. Experimental Verification. . . . . . . . . . . . . . . . . . . C. Comparison of Contrast Transfer EYciency Among Various Schemes . . . 1. Linear Forward Theory of Contrast Transfer in TEM . . . . . . . .

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69 ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(05)38002-5

Copyright 2005, Elsevier Inc. All rights reserved.

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Linear Inverse Theory for TEM Observation . . . . . . . . . . Generalized Inverse Matrix for TEM Observation . . . . . . . . Information Transfer Reliability Derived from Model Resolution Matrix Image Simulations of High‐Potential Sulfur Protein for Four Observation Schemes . . . . . . . . . . . . . . . . . . 6. Wiener Filter-Based TEM Images and Their ITRs . . . . . . . . VIII. Discussion . . . . . . . . . . . . . . . . . . . . . . . . A. Issue of Electron Loss by the Phase Plate . . . . . . . . . . . . B. Issue of Weak Objects . . . . . . . . . . . . . . . . . . . C. Issue of Specimen Charging . . . . . . . . . . . . . . . . . IX. Conclusions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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I. INTRODUCTION A. Phase Objects Transmission electron microscopy (TEM) technique is now widely used to observe various phenomena in the nanometer scale world composed of atoms and molecules. The reason TEM can enlarge such tiny materials as microscopic images is now understood by the wave optics that was originally developed in light microscopy (LM). LM uses visible light, for example, with wavelengths raging from 400 to 650 nm. On the other hand, TEM uses electron waves with wavelengths from 0.004 (
100 kV) to 0.001 nm (
1000 kV). From the theoretical viewpoint of microscopy, which asserts the theoretical resolution limit uniquely determined by the wavelength, wavelengths used in TEM seem to be too short, as in the applications to material or biological sciences the spatial resolution better than 0.1 nm is not usually called for. To understand this overspecification in terms of the wavelength limit in microscopy, we have to understand image formation and the associated contrast theory developed particularly for TEM based on the immortal work by Scherzer (1949). However, in this section, I take an approach diVerent from the authentic one to extract directly the significant role of phases in TEM (the formal theory is to be developed in Section VII.A). First, the diVraction (scattering) of electron waves by objects has to be formulated. The scattering is a summation of potential scatterings from constituent atoms in objects. A quantum mechanical theory can completely describe the scattering, but for the theory of image formation, it is enough for us to understand the electron scattering eVect by the term of space‐ dependent refractive index n(r), where the space is assumed to be two‐ dimensional. The phenomenological parameter n(r) is thought to be a sort of projection of the distribution of electric potentials of constituent atoms to

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a plane immediately after the object, which is called the exit plane. Of significance here is no absorption in this process as easily understand as no loss in electrons in scattering. Then for thin objects with a uniform thickness of ‘ the object‐dependent phase retardance is expressed as yðrÞ ¼ 2pðnðrÞ  1Þ‘=l:

ð1Þ

The term n(r)  1 corresponds to the refractive index diVerence from the vacuum state and l corresponds to the wavelength. This phase retardance occurs during the through‐object penetration of the incident wave, which is added to the original phase ot  kz z (o: temporal frequency, kz: z‐component of the wave vector k, and z is the position of the exit plane, which is perpendicular to the incidence direction). The process is schematically written: phase object

eiðotkz zÞ !

yðrÞ

! eiðotkz zþyðrÞÞ  eiyðrÞ eiðotkz zÞ :

ð2Þ

To emphasize no change other than phases as shown in Eq. (2), the object concerned is called the phase object. The transparency of phase objects is recognized as no change in the intensity of waves (the wave magnitude) before and after the object in the penetration. The fact is mathematically expressed as jeiyðrÞ eiðotkz zÞ j2 ¼ jeiðotkz zÞ j2 ¼ 1:

ð3Þ

The image formation with microscopes is a process to transfer the kind of optical disturbance by objects, eiy(r), to a distant plane (image plane) as optical images. Central in the image formation of phase objects is the recovery of the optical information, y(r), as precise as possible. We term this phase recovery, and the contrast, which represents the phase, y(r), as a magnitude, phase contrast. To do so, various optical steps such as optical conversions with lenses are devised. Among them the final step (called the image detection) is problematic from the viewpoint of the phase recovery because no change in magnitude as described in Eq. (3) simply results in no image. This is another example of a ‘‘phase problem’’ widely known in optics. There must be some innovation to overcome this fundamental issue engaging against TEM. B. A Trick to Convert Phases to Magnitudes The phase problem memtioned is quite universal in optics. It appears with diVerent faces depending on optical phenomena studied. As mentioned, in microscopy it is the problem of how to recover y(r) as an observable form or in a more straightforward term how to observe transparent objects as

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microscopic images. Zernike (1942) was one of the innovators who could give a solution to the case of visible LM, and Scherzer (1949) was another who gave a solution to TEM. Looking back to these two solutions, we see an eventual complementarity. It is the way of appearance of the lens‐dependent phase retardance, g(k), which is defind in the diVraction (Fourier) space of the objective lens and additional to the object‐dependent phase retardance y(r) through the lens aberrations and defocus (refer to Sections IV.A and VII.A for the full description of g(k) and the related subjects). To intuitively describe their complementarity in terms of contrast transfer in image formation, I will illustrate one of the approaches to address the issue of phase contrast, namely conversion of phases into the observable magnitudes. Let us restrict our intuitive argument within the use of real numbers. A simplification for the intensity detection (Eq. (3)) is also assumed as jeiyðrÞ eiðotkz zÞ j2 ! ðcosðyðrÞ þ ot  kz zÞÞ2 ;

ð4Þ 2

where the intensity detection represented by the absolute square, | | , is replaced to the time averaging of the square of consine term, cos2 ð According to this simplification, Eq. (3) reads

Þ.

1 ð5Þ cosðot  kz zÞ2 ¼ cosðyðrÞ þ ot  kz zÞ2 ¼ : 2 In this formulation, the above equation is also showing no way to observe the phase term y(r). I introduce a following trick to extract the term y(r) even with this intensity detection. cosðyðrÞ þ ot  kz zÞ ¼ cosyðrÞcosðot  kz zÞ  sinyðrÞsinðot  kz zÞ # trick fðr; tÞ ¼ cosyðrÞcosðwt  kz zÞ  sinyðrÞcosðot  kz zÞ

ð6Þ ð60 Þ

Let us detect the wave converted by the above trick: fðr; tÞ2 ¼ cosðot  kz zÞ2 ðcosyðrÞ  sinyðrÞÞ2 1 ¼ ð1  2cosyðrÞsinyðrÞÞ 2 1 ¼ ð1  sin2yðrÞÞ 2

ð7Þ

Particularly when y(r) 1, which is the bright field condition and inevitable in the linear extraction of yðrÞ; fðr; tÞ2 is approximated as fðr; tÞ2

1  yðrÞ: 2

ð8Þ

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The observed image fðr; tÞ2 includes y(r) as a spatial change in magnitude, namely a phase contrast. We can materialize the mathematical trick given by Eqs. (6) and (60 ) by using the Fourier transform (FT) characteristics of microscopy and the additional lens‐dependent phase retardance g(k). The condition y(r) 1 simplifies Eq. (6) as
 cos yðrÞ þ ot cos ot  sin otyðrÞ; ð9Þ where term kzz is ignored because it always appears as a constant term. The diVraction (Fourier) image obtained at the back‐focal plane (k space) of the objective lens becomes FT½cos ot  sin otyðrÞ ¼ cos otdðkÞ  sin ot~yðkÞ;

ð10Þ

where FT [ ] represents the Fourier transformation, d(r) indicates a delta function representing a sharp focus of the parallel direct beam at the back‐ focal plane, and ~ yðkÞ is the Fourier transform (diVraction) of y(r). Generally, the phase retardance g(k) is stepped into the phase term as,

 cos ot þ gðkÞ dðkÞ  sin ot þ gðkÞ ~yðkÞ; ð11Þ as g(0) ¼ 0 and d(k) ¼ 0 at |k| 6¼ 0, above equation is further simplified.
 cos otdðkÞ  ~ yðkÞsin ot þ gðkÞ
 ð12Þ ¼ dðkÞ  ~ yðkÞsingðkÞ cos ot  ~ yðkÞcos gðkÞsin ot: At the image plane, we have a wave function that is the inverse Fourier transform of Eq. (12).
 1  yðrÞ  FT 1 ½singðkÞ cos ot  yðrÞ  FT 1 ½cos gðkÞsin ot ð13Þ Here, FT 1[ ] represents the inverse Fourier transformation. When we insert a Zernike phase plate into the back‐focal plane, d(k)cos ot is left unchanged and the term with ~ yðkÞ shifts its phase by p/2. Eq. (12) is converted to
p dðkÞcos ot  ~ yðkÞsin ot þ gðkÞ  ð14Þ 2 ¼ bdðrÞ  ~ yðkÞcos gðkÞccos ot þ ~ yðkÞsin gðkÞsin ot: Then the wave function at the image plane becomes 1  yðrÞ  FT 1 ½cos gðkÞcos ot þ yðrÞ  FT 1 ½sin gðkÞsin ot:

ð15Þ

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Images corresponding to the conventional and the Zernike can be obtained as the square of Eqs. (13) and (15): 1 Conventional : ½1  2yðrÞ  FT 1 ½singðkÞ 2

ð16Þ

1 Zernike : ½1  2yðrÞ  FT 1 ½cosgðkÞ; 2

ð17Þ

where the square terms proportional to y(r)2 are ignored. In the traditional formulation of contrast transfer, the modulated phase contrast, yðrÞ  FT 1 ½singðkÞ or yðrÞ  FT 1 ½cosgðkÞ, is interpreted as the interference of primary waves directly through objects and the waves scattered by objects, which is described in detail in Section VII.A. In the above formulation, so‐called contrast transfer functions (CTFs), sing(k) and cosg(k), linearly contribute to the final images. The type of functional forms, sine or cosine, is crucial to determine the contrast of images, which is governed by the behavior of low‐frequency components in images. One can adjust the defocus to recover the contrast through sing(k) for the conventional (the defocus contrast for the conventional TEM), but the contrast given through sing(k) is always much smaller than the contrast given through cosg(k), which is the case for the Zernike at low frequencies. At higher frequencies, on the other hand, the sin g(k) CTF must be much preferred. Their complementarity has long been recognized in visible LM, and the high contrast images for phase objects obtained by Zernike phase contrast microscopes have greatly contributed to biology. Unfortunately in TEM, however, only the conventional and the sine‐type contrast transfer has been appreciated. II. ISSUES in PHASE RECOVERY in TEM A. Historical The remarkable merit to use electrons as the information carrier or the atomic probe for materials is transformed into a demonish demerit when materials are unexpectedly charged with the electron irradiation itself. Historically, we could see various examples where the charging issue is so severe that one had to give up theoretically excellent ideas. In the beginning of the TEM innovation, the demon appeared as trouble in using static lenses or foil lenses. For the application of SEM to biological materials, charging of objects has always been the fundamental obstacle for obtaining correct

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images. The same was true for the phase plate. The static charges induced somewhere in the phase plate by electrons, of which major cause might be the electron‐inducing ionization of atoms, behave as microlenses that lead inevitably to hectic deterioration of resultant images. Studies conducted by many forerunners to import the Zernike idea to TEM had always met this bottleneck and been forced to give up (Badde et al., 1970; Balossier et al., 1981; Boersch, 1947; Faget et al., 1962; Johnson et al., 1973; Kanaya et al., 1958; Krakow et al., 1975; Unwin, 1970; Willasch, 1975). After the beginning of the 1980s, no challenge has been made for the development of the phase contrast TEM with phase plates. General understanding in the society of electron microscopy seemed to be the acceptance of the defocus phase contrast (DPC) complementary to the Zernike phase contrast (ZPC), as memtioned. To compensate the weakness inherent in the defocus method without dehancing the image contrast, various ideas have been put forth such as deep defocusing, defocus variation, and Wiener filters. A major focus common to the improvements is how to recover the contrast governed by low‐frequency components which are severely surpressed by the kind of CTF, sin g(k) (Reimer, 1997). Actually there has been great success in the TEM application to material sciences in conjuncture with the theoretical approach enhanced with computer simulations (Peng et al., 2004). But this is feasible only when a high dose can be accepted to objects. We have had a great amount of trouble, however, in using high dose for biological samples due to the easy degradation of organic materials by electron bombardment. Intrinsically low eYciency in electron scattering by light elements and the dose limitation mentioned have been making it almost impossible directly to observe biological samples without heavy element staining. This is the place where the Zernike method comes into play, with an expectation to enhance the image contrast high enough to work on biological samples without staining. All of the past eVorts, however, did not invite successful results. Therefore, the major issue in the development of phase plate TEM is how to solve the phase plate charging. B. Amount of Charges in the Phase Plate Where are charges in the phase plate? It has been diYcult to answer the question, as there is no direct way to visualize charges as images. What is the charge origin? This is also diYcult to answer because a countable number of charges are enough to destroy the contrast transfer process in TEM, as found in our study (Danov et al., 2001, 2002). Figure 1 shows how free charges can extend their influence to the free space as an electric potential. The integration of this potential along the electron path gives an additional

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FIGURE 1. Typical two‐dimensional plot of the dimensionless electrostatic potential induced by a circular plate possessing a uniform positive charge (from Figure 2 in Danov et al., 2001).

phase retardance to electron waves. The charge‐inducing phase retardance is enormous even if the amount of charges is very little, say, only 10 elementary charges on one phase plate, as shown in Figure 1. Conversely, the charge‐inducing surface potential in the phase plate can be obtained by the comparison of CTFs with and without the phase plate. Such an example is shown in Figure 2, where the surface potentials were obtained for a uniformly thick carbon film with a combination of the CTF measurement and computational calculation based on a pertinent theoretical model (Danov et al., 2001). The potential of about 0.3 V resulted for two experiments under the assumption of a uniform surface potential for the phase plate corresponds to about 10 elementary charges in total, uniformly spreading in the phase plate in an averaged sense. A more elaborated approach, which can estimate the distribution of charge density on the phase plate, has also resulted in a countable elementary charge suYcient to induce phase disturbances to the lens system (Danov et al., 2002). This result is telling the charging is the fundamental issue with phase plates.

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FIGURE 2. Experimental evaluation for the surface potential (V0) (from Figures 7 and 9 in Danov et al., 2001). (a) Contrast transfer functions of two images taken at same defocus with and without a uniform carbon film in the back‐focal plane. The curves are displaced vertically for better viewing. (b) Experimental results for the phase shifts due to charging (symbols; □ and ) and best theoretical fits (lines). The results are from two experiments (A and B).

▪

C. Remedies for Phase Plate Charging For several years of our struggle for the settlement of the charging issue, the first few years focused on specifying the charge origin. As many forerunners have pointed out, the surface contamination with insultating materials was immediately found to be the source. The phase plate itself is not charged when it is made of conducting material such as carbon. The identification of

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the type of contamination materials was rather a diYcult task. In lengthy cut‐and‐try procedures such as the variation of phase plate materials, heating the phase plate, surface cleaning with acid or organic solvent linsing, ion sputtering or oxygen etching, we have concluded that there are three sources for the charge contamination: organic materials, metal oxides, and inorganic materials. After this identification, studies on the specification where and when those contamination materials have stepped into the phase plate surface followed. The result of source hunting is summarized in Table 1. Organic molecules have long been recognized as the major source of charge contamination, and several ideas have been proposed to remove the contamination, which include linsing with organic solvents, heating, ultraviolet (UV) cleaning, ion sputtering, or electron‐preirradiation of the phase plate. Among these remedies, the heating seems to be most eVective, but the organic‐free preparation of phase plates is only half of the solution because originally clean phase plates are easily contaminated inside the TEM column with organic molecules evaporated from biological samples themselves once bombarded by electrons. Therefore, continuous heating of phase plates inside the column must be requested, as suggested decades ago (Sieber, 1974). Metal oxides are very severe sources for charging. This is the major reason we have to stick to carbon as the material for phase plates. During the deposition process of carbon films for phase plates, for example, with a vacuum evaporator, the metal oxide contamination must be carefully avoided by using metal oxide-free devices. We have found the best materials for devices are glass, stainless steel, and carbon. TABLE 1 SOURCE SPECIFICATION OF CONTAMINATION‐INDUCING CHARGES IN PHASE PLATES Origins Organic materials

Metal oxides

Inorganic materials

a

Sources * * * * *

Backflow of oil mist from vacuum pumps Silicone grease inside the TEM column In situ contamination from organic specimens Surface oxidation of phase plates made of metalsa Gallium oxide deposited to the phase plate during the focused ion beam fabrication * The sputter from metal walls of device parts inside the vacuum evaporator * Mica flakes adhered from the mica plate during the exfoliation * Salt deposition from the rosksalt during the exfoliation on the water surface

The major reason we have to stick to carbon as the material for phase plates.

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Inorganic contamination, particularly from the mica surface, with such an amount as to become the source of countable charges is almost impossible to be avoided. We have had to severely think about this because the exfoliation of carbon films from the mica surface must reflect a nature of multilayerd 2D crystalline mica. The separation crack may mostly come to the mica layers gap but not to the gap between the carbon film and the mica, which naturally leads to the attaching of mica flakes to the carbon film. Thus, our charge origin hunting had concluded such a pessimism that there was no way to get rid of the charge contamination in the procedure of the phase plate production. We encountered a situation to give up the thorough cleaning of phase plates and jump to completely diVerent directions. In short, we had to find a magic to kill the charge eVect without removing charges. If we could recognize the physical cause behind the pathology as the long range potential induced by free charges as described but not the charges themselves, we might find the remedy. It took still a few years, however, to find an actual remedy even after we could fix charge origins. To solve the phase plate issue, we have finally come to the conclusion that we must employ an electrostatic shield of the charge‐inducing potential by coating with conductive materials, here again, carbon. The essence of the remedy is the recognition that the existence of charges and the charge eVect are separable. In the ultimately final step for the phase plate production, both sides of the phase plate seemingly carrying organic materials, metal oxides, or inorganic materials were carbon coated with a vacuum evaporator. The coated carbon wrapping the whole phase plate must stay clean and, once grounded, kill the charge potential. The schematics of a three‐layered phase plate made of carbon is shown in Figure 3b together with a conventional Zernike phase plate for comparison (Figure 3a) (Nagayama et al., 2004b). Carbon films thicker than 5 nm were found to be eVective enough to shield the electrostatic potential and recover the CTF as theoretically expected. To what extent the charge‐dependent phase retardance additional to the p/2 shift of the Zernike phase plate is surpressed with the two‐sided carbon coating is detailed in Section IV.C.

D. Spatial Filters for Phase Contrast Once the charging issue is solved, we can approach the problem of phase contrast from a diVerent angle. The notion of phase plates can be extended to cover various spatial filters, which can manipulate optical information in the Fourier space (refer to Figure 29 in Section VI for the mathematical formulation of filters). Many ideas could be borrowed from the extensive study conducted once in classic optics (Wolter, 1956). The oldest kind of

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FIGURE 3. A remedy to fix the charging problem in the phase contrast TEM with phase plates (from Nagayama et al., 2004b). (a) Typical design of the Zernike phase plate made of carbon for the acceleration voltage of 120 kV. (b) Three‐layered carbon film designed for the Zernike phase plate to avoid the charging eVect. The most distinctive in making the layered structure is the carbon‐coating, which has to be done in the final stage in the whole procedure.

spatial filters is the Foucault knife edge, which was invented in the middle of the nineteenth century by Foucault and is still used in the field of photoimaging as the Schlieren optics (see Figure 29). As described in Section VI, the function of the knife edge can be decomposed to a summation of two basic functions: an identity function, 1, associated with no filter and a signum function, sgn x, associated with a half‐plane p phase filter. Actually, the half‐plane p phase filter itself was tested in the visible light optics to observe phase objects (Lowenthal et al., 1967). Unfortunately, its potential to mimic the diVerential interference contrast, which is the core of our finding bearing fruit as the Hilbert diVerential, has long been overlooked. The reason is discussed in Section V. In this chapter, three kinds of phase plate TEM using three spatial filters are described. Their designs are shown in Figure 4, together with the conventional TEM design. The back‐focal plane after the objective lens is the most significant moment to be managed in these designs. Conventionally in the back‐focal plane, there is an aperture to control the frequency components involved in images by limiting the highest usable frequency. The hole size of the aperture is adjusted to range from 5 to 100 mm, for example. The phase plate made of carbon film is placed on the aperture.

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FIGURE 4. Phase contrast schemes using phase plates as spatial filters. (a) Aperture‐only design where phase contrast is recovered by the control of the defocus. (b) Zernike phase plate filter where contrast is recovered under the near‐focus condition. (c) Half‐plane phase plate filter where Hilbert differential phase contrast is recovered under the near‐focus condition. (d) Foucault knife‐edge scanning filter where Foucault differential phase contrast is recovered even for strong objects.

For conventional TEM, nothing is placed on the aperture (Figure 4a). For the Zernike phase contrast, a p/2‐phase plate with a tiny hole in the center is placed, which exclusively manipulates scattered waves by adding a phase of p/2 (Figure 4b). The filter functions to convert the scattered wave front (WF) C (Figure 4a) to iC (Figure 4b). For the Hilbert diVerential contrast, a p phase plate with a hemicircular shape is placed on the aperture, which negates the half part of the scattered WF from C to C (Figure 4c). For the Foucault diVerential contrast, a Foucault knife edge is placed and scanned

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from the completely open to the completely closed state on the aperture (refer to C and 0 in Figure 4d). According to the interception of electron waves with these phase plates, WFs transmitted through the microscope are manipulated in their frequency components, namely filtered. To understand the function of filters shown in Figure 4, we summarize their function forms in Table 2, together with their images. In Table 2, weak objects ðy2 ðrÞ jyðrÞj 1 and then eiyðrÞ 1 þ iyðrÞÞ are assumed for the conventional, the Zernike, and the Hilbert, but no assumption is made for the Foucault. Comparison among the four filters immediately tells us that the conventional image must bear only low contrast because no term linear to y(r) exists. The y(r) terms observed in the other three schemes have their own characteristics, which might reflect in the image appearance, as illustrated in the following sections. TABLE 2 FILTER FUNCTIONS ASSOCIATED WITH PHASE PLATES AND RESULTANT IMAGES

Schemesa

WF at the exit planeb

Filter functionc

WF at the image planed

Image with intensity detectione

Conventional (defocus phase contrast)

1 þ iy(r)

1 þ iy(r)

1 þ y(r)2

Zernike phase contrast

1 þ iy(r)

1 þ y(r)

1 þ 2y(r) þ y(r)2

Hilbert diVerential contrast

1 þ iy(r)

1 1 þ px  yðrÞ

2 1 þ px y(r) 1 þ ð px  yðrÞÞ2

Foucault diVerential contrast

eiy(r)

  iyðrÞ d 1  2pi dx e

1 dyðrÞ 1 þ px
dx 2 1 dyðrÞ þ 2p dx

a

Detailed descriptions on the Zernike, Hilbert, and Foucault schemes are followed in Sections IV, V, and VI. b 1, incident or primary wave; iy(r), scattered wave; eiy(r), total wave. c Idealized filters, which do work without the loss of electrons, assumed. Solid line, real filter; broken like, imaginary filter; |k|, wave vector magnitude or radial component of wave vector. d 1 CTF modulation is ignored for simplification (refer to Table 5 for CTFs). px  C, Hilbert d transform; *, convolution; dx , derivative in the x direction. e Squared WF.

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III. DEDICATED PHASE PLATE TEM In this section, a phase plate equipping system and its application to 120‐kV TEM is reported (Hosokawa et al., 2005). The TEM was modified with an additional lens doublet, which allowed more flexibility in applying phase plates. The doublet transfers the image from the back‐focal plane to a plane far below the objective lens where a special phase plate holder, anticontaminator, and other necessary devices could be employed. Using a heating holder to protect the phase plate from the charge contamination was investigated. A. Transfer Lens Doublet The phase plate needs to be placed at the back‐focal plane. In conventional TEM, the space around the back‐focal plane is very limited because it is situated close to the specimen in the pole piece gap. Then possibilities for the resolution of charge contamination issue must be limited. The movement mechanism of the conventional aperture holder is also lacking in the required precision for aligning the phase plate on the optical axis. Figure 5 shows the technical solution employed in phase plate TEM to provide a large free space for the phase plate holder. This solution can preserve the conventional optical properties of the TEM. A confocal doublet placed after the objective lens transferred the back‐focal plane onto the phase plate plane located in mechanically free space. This doublet ‘‘clones’’ electron waves from the back‐focal plane onto the phase plate plane preserving both the position and the slope of the first‐order electron trajectories. The other lenses function in the same manner for the various imaging modes as in conventional TEM. The image after the phase plate can be formed in suYciently high magnification, and the image is not degraded due to aberrations of the intermediate lenses. After accounting for the focal length of the objective lens (1.7 mm) and the focal length of the transfer doublet (42 mm), the admixture of aberrations introduced by the transfer doublet can be estimated (Hosokawa et al., 2005). For example, the third‐order spherical aberration constant Cs defined at the center of the two lenses is converted to the DCs at the specimen plane by Eq. (18). DCs ¼ ð1:7=42Þ4 Cs

ð18Þ

which is negligibly small. Because the doublet is a weak lens, the chromatic aberration constant can be taken as equal to the focal length. The augmentation, DCc, of the chromatic aberration constant at the object plane due to the transfer doublet was estimated also to be negligible (Hosokawa et al., 2005).

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FIGURE 5. (a) A whole view of the complex part consisted of the specimen holder, the transfer lens doublet, and the phase plate holder (from Figure 1 in Hosokawa et al., 2005). (b) Diagram of the transfer lens doublet. The back‐focal plane and phase plate plane are optically equivalent in the first‐order trajectory. Two deflectors (Def1, Def2) are used to align the beam in the doublet, and one (Def3) for the image shift. A conventional specimen holder is used as the phase plate holder.

Table 3 shows the numerical calculation results of the spherical and chromatic aberration constant using the axial magnetic flux. The anisotropic oV‐axial aberrations such as coma and anisotropic chromatic aberration can degrade the image at lower magnifications. Exciting the two lenses with opposite polarity can cancel these aberrations. To align the electron beam into the transfer doublet, two deflection coils were located above each lens of the doublet (Figure 5). Another deflection coil was provided above the phase plate. It can be used for the ‘‘image shift’’ similar to conventional TEM. A conventional specimen stage could be employed as phase plate stage. This high‐precision positioning system easily and accurately aligned the phase plate. B. Phase Plate Holder The phase plate was supported by a copper grid, which is typically used for specimen support. The grid was placed on a conventional specimen holder.

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PHASE CONTRAST ENHANCEMENT TABLE 3 SPECIFICATIONS OF THE DEDICATED PHASE PLATE TEM (FROM TABLE 1 HOSOKAWA ET AL., 2005)

IN

Electron source

LaB6

Accelerating voltage Objective lens

120 kV Cs ¼ 1.0 mm Cc ¼ 1.3 mm f0 ¼ 1.7 mm Amorphous carbon Thickness: 23 nm Hole diameter: 1 mm Cs ¼ 4400 mm (0.012 mm at object plane) Cc ¼ 78 mm (0.15 mm at object plane) f ¼ 42 mm

Phase plate

Doublet

Cs, spherical aberration constant; Cc, chromatic aberration constant; f, focal length.

One advantage of using a standard specimen holder and mechanical stage was that they are already developed and thoroughly tested. The precise mechanical positioning capability of the stage easily aligned the phase plate. First, the doublet is slightly defocused, which produced a ‘‘shadow image’’ of the phase plate where the phase plate hole is clearly visible on the screen. Mechanically moving the phase plate centered the hole on the screen. Then focusing the doublet positioned the diVraction crossover inside the phase plate hole. This was observed as an infinite enlargement of the phase plate hole shadow on the screen. Besides the ease of alignment, the specimen holder can be heated. This way, the phase plate was maintained at suYciently high temperatures inside the microscope, which can avoid the charge contamination from organic materials as described. Figure 6 shows the heating holder used in our experiments. The specifications are: electric heater, a variable temperature range from room temperature to 1000  C, and a thermocouple for measuring the temperature. A liquid nitrogen anti‐contamination cooling trap was also mounted around the phase plate. C. EVect of the Phase Plate Heating Holder Figure 7 clearly demonstrates the importance of using a heating holder for phase plate support. Figure 7a shows a TEM image of the Zernike phase plate after being used for several days of TEM experiments when heating the phase plate to 200  C. Despite the large defocus, contaminants were not

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FIGURE 6. A whole view and a closeup view of a phase plate holder (from Figure 2 in Hosokawa et al., 2005). The phase plate is heated while inside the microscope using the heating holder to avoid contamination of the phase plate surface during experiments.

FIGURE 7. TEM images of the Zernike phase plate after it has been used for some time (from Figure 6 in Hosokawa et al., 2005). (a) The heating holder maintained the phase plate at 200  C during several days of experiments. (b) The phase plate used at room temperature for several hours of experiments.

observed and the phase plate looked clean. Figure 7b shows the same phase plate after it was used at room temperature for several hours of experiments. Many spots and darker areas were observed around the hole, which resulted from electron beam-induced contamination. Thus, using heat is essential for avoiding in situ organic contamination on phase plates inside TEM.

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IV. ZERNIKE PHASE CONTRAST TEM The possibility of implementing a Zernike phase plate into a transmission electron microscope is investigated both theoretically and experimentally (Danev et al., 2001b). The phase‐retarding plate in the form of thin film with a hole in the center covers the aperture positioned in the back‐ focal plane. The experiments show that the phase plate functions as predicted, a cosine‐type phase contrast transfer function (cos‐CTF). Images of negatively stained horse spleen ferritin were highly improved in the contrast and the image modulation, compared to those acquired without the phase plate. A. Contrast Transfer As explained in Section I.B, in a weak phase object approximation, the CTF of the microscope is of the form GC ðkÞ ¼ singðkÞ:

ð19Þ

The CTF phase retardance g(k) is due to the defocusing and the spherical aberration of the objective lens and is given by the formula   1 1 3 4 2 gðkÞ ¼ 2p  Dzlk þ CS l k ; ð20Þ 2 4 where Dz is the defocus, CS is the spherical aberration coeYcient of the objective lens (refer to Section VII.A for details). The introduction of a phase plate with a thickness for the ’0 ¼ p/2 phase shift modifies the CTF to  singðkÞ; k < kh ; ð21Þ Gz ¼ cosgðkÞ; k  kh where kh is the wave number corresponding to the edge of the central hole of the Zernike phase plate. The so‐called Scherzer defocus, defined as the value for which the contrast of a point phase object is maximum, is given for the defocus phase contrast (DPC) (Danev et al., 2001): pffiffiffiffiffiffiffiffiffi DzC ¼ 1:21 CS l ð22Þ The corresponding parameter in the case of the Zernike phase contrast (ZPC)—the defocus for which the contrast of a ZPC point phase object is maximum—is given by

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DzP ¼ 0:73

pffiffiffiffiffiffiffiffiffi CS l:

ð23Þ

The above values are calculated for each aperture cutoV frequency coinciding with the first intercept between the CTF and the frequency axis. These cutoV frequencies, in the case of optimal defocus, are kC ¼

1:56 ðCS l3 Þ1=4

ð24Þ

and kP ¼

1:40 ðCS l3 Þ1=4

ð25Þ

for DPC and ZPC, respectively. The resolution determined by the cutoV is 11% higher in DPC when compared to ZPC. Figure 8 shows plots of CTFs at optimal defocus. Although the corresponding ZPC has a lower cutoV frequency, it preserves the lower part of the frequency spectrum much better. In order to achieve similar low‐ frequency performance, DPC‐TEM needs strong defocusing, which leads to a vast reduction in the direct resolution. The low‐end limit of the cos‐CTF

FIGURE 8. Plots illustrating the CTFs for DPC‐ and ZPC‐TEMs as a function of the 3 1=4 ^ reduced spatial frequency pffiffiffiffiffiffiffiffi k ¼ kðCs l Þ (from Figure 2 in Danev et al., 2001b). The reduced defocus ðD^z ¼ Dz= Cs lÞ values, Dz^¼ 1.21 for DPC‐TEM and Dz^¼ 0.73 for the ZPC‐TEM, are calculated for the optimal contrast of a point phase object. A partial sin‐CTF, at Dz^ ¼ 8.0, is shown as an illustration of the deeper focus in DPC‐TEM.

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passband, given in Eq. (21), is not illustrated in the figure (infinitely small hole approximation). B. Optimum Material for Phase Plates The presence of material film in the back‐focal plane of the objective will result in scattering of some of the information‐carrying electrons and therefore reduce the signal‐to‐noise ratio in the image. In Table 4 are shown the calculated values for the p/2 thickness of diVerent elements at two acceleration voltages (100 and 300 kV) (Danev et al., 2001b). The inner potential values are taken from Reimer (1997). For those thicknesses, the number of unscattered electrons (electron transmittance) was calculated using an empirical model (Reimer, 1997) and is illustrated in Figure 10. The results for carbon are in good agreement with experimentally measured values (Angert et al., 1996; Sugiyama et al., 1984). For the lighter elements (z < 20), the inelastic scattering dominates over the elastic. The increase of the accelerating voltage decreases the total number of scattered electrons, but the values stabilize, and above 300 kV, much improvement was not observed. The lighter elements (Be, Al, and Si) show the lowest total scattering, so they should be considered as main candidates for phase plate preparation. Beryllium was previously proposed for the purpose of phase plate preparation (Badde et al., 1970), but its high toxicity makes it inappropriate from a practical point of view. Carbon, though showing about 10% higher

TABLE 4 CALCULATED THICKNESS FOR p/2 PHASE SHIFT FOR FILMS MADE OF DIFFERENT ELEMENTS (FROM TABLE 1 IN DANEV ET AL., 2001b)  p/2 thickness (nm) Element

Inner potentiala (V)

100 kVb

300 kVb

7.8 7.8 12.1 11.5 20.1 15.6 20.7 23.4 21.1

21.79 21.79 14.04 14.78 8.45 10.89 8.21 7.26 8.05

30.86 30.86 19.89 20.93 11.97 15.43 11.63 10.29 11.41

4Be 6C 13Al 14Si 29Cu 32Ge 47Ag 74W 79Au a

Values taken from the literature (Reimer, 1997, p. 51).

b

Acceleration voltage.

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FIGURE 9. Elastic, inelastic, and total electron transmittance calculated for films whose thickness is such as to obtain a p/2 phase shift (from Figure 3 in Danev et al., 2001b). The chemical element symbols are indicated on the horizontal axis. The graph has two sets of plots: one for 100‐kV and the other for 300‐kV acceleration voltage.

scattering than Al and Si, has the practical advantage of easy preparation and conductive non-metal. In the worst case, every information‐carrying electron that has been scattered from the plate (elastically or inelastically) is prevented from contributing to the phase image. Such electrons are added to the background of the image. Implementation of energy filtering to remove inelastic scatterings from the object and the phase plate will improve the performance of ZPC‐ TEM. The ratio unscattered/incident number of electrons may be considered the ‘‘transparency’’ of the phase plate. If the plate has a uniform thickness, the signal reduction ratio will be constant throughout the spectrum. The data in Figure 9 are a rough estimation used to compare diVerent materials. C. Comparison Between Zernike Phase Contrast and Defocus Phase Contrast Figure 10 shows the moduli of the Fourier transforms of images taken at same defocus (830 nm, overfocus) with and without the phase plate. For the specimen, thin amorphous carbon film was used as a weak phase object. Figure 11a shows the rotationally averaged and background subtracted profiles of the CTF patterns from Figure 10a and b. A rough comparison of the CTF amplitudes shows about 20% signal reduction in the cosine‐type image. The phase plate produces cos‐CTF. Theoretically, the sine type and

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FIGURE 10. Diffractograms of images taken at same defocus (830 nm) without ( (a) DPC‐ TEM) and with ( (b) ZPC‐TEM) a phase plate in the back‐focal plane (from Figure 5 in Danev et al., 2001b).

FIGURE 11. CTF phase evaluated for a conventional Zernike phase plate. Experiments were made with a 300‐kV TEM (from Figure 5 in Danev et al., 2001b). (a) Rotationally averaged profiles of the diffractograms (Figure 10a and b). (b) CTF phase, extracted by fitting the extrema positions in the profiles in a. The difference between the two CTF phases (○: without phase plate, : with phase plate) gives the phase retardance ( ) introduced by the phase plate.

▪

▴

the cos‐CTF should be complementary (i.e., the intercept of the conventional sin‐CTF with the frequency axis should coincide with the maxima of the cosine‐type CTF, and vice versa). However, due to the charging, the cos‐ CTF is distorted. Figure 11b shows plots of the phases extracted from the CTFs in Figure 10a, by fitting the extrema positions. The phase of the sin‐ CTF matches the theoretical model (Eq. (20), Dz ¼ 830 nm, CS ¼ 3mm, solid line). The interpolated values for the sin‐CTF are then subtracted from the cos‐CTF, leaving only the phase retardance due to the phase plate. Ideally, the phase shift introduced by the phase plate should be a flat curve. In this case, however, it starts with a negative value (0.4p at k ¼ 0.5 nm1)

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FIGURE 12. CTF phase evaluated for a three‐layered (anticharge) Zernike phase plate with a dedicated phase plate TEM (120 kV) (from Nagayama et al., 2004b). (a) Rotationally averaged profiles of diffractograms (not shown). (b) CTF phase, extracted by fitting the extrema positions in the profiles in a. The difference between the two CTF phases (○: without phase plate, : with phase plate) gives the phase retardance ( ) introduced by the phase plate.

▪

▴

and gradually increases with the increase of k. This behavior is similar to overfocusing and is explained by the presence of positive charges around the center of the phase plate. The phase delay caused by the material film itself cannot be precisely determined from these data. Subjective extrapolation k ! 0 shows that it is slightly below 0.5p. To avoid the phase plate charging, we applied the three‐layered carbon phase plate, as explained in Section II.C, to the Zernike phase plate. Once the phase plate is contaminated by inorganic materials or metal oxides, it cannot be removed simply by heating. Therefore, we have to prepare the contamination‐free phase plate from the beginning before its insertion into the TEM column. The surface of three‐layered carbon films can be made contamination free owing to the specific fabrication procedure. Our plan has been proven with satisfaction, as shown in Figure 12. Figure 12a is the rotationally averaged and background‐subtracted profiles of the CTF moduli obtained for a thin amorphous carbon film (diVerent from the one shown in Figure 10) with and without the three‐layered phase plate. Figure 12b shows plots of the phases extracted from Figure 12a. Both CTFs, the sine type and the cosine type, make an almost constant diVerence about p/2 for the phase plate-dependent phase retardance. Judging from Figure 12b, this charge eVect–free phase plate may guarantee the right Zernike phase contrast up to the spatial frequency of 1.5 nm1. Figure 13 shows images taken from negatively stained horse spleen ferritin molecules, supported by an amorphous carbon film. The phase TEM image (Figure 13a) demonstrates that a very high contrast can be obtained with a resolution cutoV above the CCD Niquist frequency—that is, 0.41 nm. The

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FIGURE 13. 300 kV cryo‐TEM images of negatively stained horse spleen ferritin (from Figure 6 in Danev et al., 2001b). (a) ZPC‐TEM image acquired using a Zernike phase plate. (b, c, and d) Conventional TEM images at a defocus of 2550, 540, and 130 nm, respectively (underfocus). The insets show the diffractogram for each image. The scale bars in the insets correspond to 1nm1.

compromise between contrast and direct resolution is clearly illustrated by the conventional images. The image with the deepest underfocus (Figure 13b) shows a contrast comparable to that of the phase image; however, the direct resolution is limited to 2.2 nm. On the other hand, the image closest to the Scherzer defocus (Figure 13d) shows a very low contrast, although the high‐frequency components are well preserved up to 0.45 nm. D. Biological Applications As long experienced, biological samples without heavy element staining exhibit very weak contrast. To enhance contrast, two methods are typically employed: (1) scattering contrast with small aperture diaphragms and (2)

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DPC with deep defocusing. The former technique is used in histochemical sciences, where sectioned samples of cells or tissues are imaged, and the latter is popular in electron crystallography, where very thin samples are used. In both methods, however, the contrast is gained by impairing the intrinsically very high resolution of the TEM. This can be overcome through the combined use of ZPC‐TEM and rapid freezing technique. Rapidly frozen ice‐ embedded specimens provide the most realistic images (Fernandez‐Moran, 1960; Heuser et al., 1979; Van Harreveld et al., 1964), as they are free from artifacts induced by sample preparation methods, such as chemical fixation, dehydration, staining, and sectioning. In our design of the Zernike phase plate, due to the finite size of the central hole (1 mm f), there is a limit in the recoverable lower spatial frequencies. That is about (0.04 nm1) for our 300‐kV TEM system, for example. Within this limitation, the most eYcient application of the ZPC‐TEM to biology is the structural study of viruses, of which geometry is around 50 nm. A few examples of our experience for the virus TEM are shown. The experiments were carried out on a JEOL JEM‐3100FFC electron microscope operated at 300‐kV acceleration voltage with or without the Zernike phase plate. The phase plates were made of an amorphous carbon film of a thickness designed to be 32 nm. The microscope was equipped with a field‐emission gun and omega‐type energy filter. Objective lens parameters were: spherical aberration coeYcient 5 mm and chromatic aberration coeYcient 4.7 mm. All observations were performed with a nominal magnification of 10,000 and an electron dose of about 100 e/nm2 in zero‐loss filtering mode. The energy window width was set at 10 eV. A special aperture holder with heating was used to support the phase plates. To avoid contamination, the phase plates were kept at approximately 200  C at all times. All images were recorded with a Gatan MegaScan795 2K  2K charge‐coupled device (CCD) camera. Ditgital Micrograph, supported by Gatan, was used for image analysis. The electron dose on the specimen could not be determined precisely, because of the scattering absorbance by the ice surrounding the specimen. The electron dose on the CCD, however, could be measured accurately as 10.5 e nm2 for DPC‐TEM and 8.1 e nm2 for the ZPC‐TEM method. The CCD dose by ZPC‐TEM was lower because of the scattering absorbance by the phase plate. Considering the position of the CCD camera, final magnification was calculated as 16,000. Since the physical pixel size of the CCD was 30 mm, the resolution was 1.78 nm pixel, which gives a Nyquist wavelength of 3.56 nm (0.28 nm1). Samples were collected by centrifugation and dropped on a copper grid coated with carbon film. After removing excess liquid carefully with the tip of a filter paper, the sample was frozen rapidly in liquid ethane using a LEICA rapid‐freezing device (LEICA EM CPC). The grid with

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ice‐embedded samples was transferred to the specimen chamber of the TEM using a cryo‐transfer system. 1. ZPC‐TEM Images of Virus Species First let us see how diVerent are the contrast of images recorded with the two conditions, with and without the Zernike phase plate. Figure 14 is showing two images for a virus, bacillus phage f29. Bacillus phage f29 was purified by centrifugation through a discontinues CsC1 gradient and suspended in 0.15 M Tris buffer containing 0.1 M NaCl and 0.01 M MgCl2 (pH7.5) (Hirokawa, 1972). At a first glance, it is very evident that the image contrast for the ZPC is quite higher than for the DPC. We can recognize such aspects as i) coat proteins and ii) head projections on the surface, all of which are easily overlooked in the DPC. Particularly interesting features observed in the ZPC are, i) two distinctive particle contrasts and ii) a portal structure on the round‐looked phages (refer to an arrow). The suspension, stocked in a refrigerator for years, was vitrified in thin layer for phase contrast transparency electron microscopy. Due to an aged suspension of phage f29, so many tail‐les and empty head of phages can be seen under electron microscopy. A small hole in the center of empty head, from the bottom view, is probably the portal in which phage DNA can get in and out for packaging and infection, respectively. We can fairly recognize the tail structure also for the hexagon‐looked viruses. Another example is the reconstituted poliovirus shown in Figure 15 (Cheng et al., 2004). In the procedure of reconstitution, the RNA molecule can be controlled to get into the virus or not for packing. The reconstituted polioviruses shown in Figure 15a have filled RNA molecules but those shown in Figure 15b have no RNA. As discussed for the image in Figure 14,

FIGURE 14. 300 kV cryo‐TEM images of a bacillus phage, f29 (from Hirokawa et al., 2004). (a) Conventional TEM image acquired without phase plate. (b) ZPC‐TEM image acquired with a Zernike phase plate.

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FIGURE 15. ZPC‐TEM images (300 kV) ice‐embedded unstained polioviruses (reconstituted sample) (from Cheng et al., 2004). (a) An image of DNA‐filled polioviruses. (b) An image of DNA‐unfilled polioviruses.

the DNA‐filled viruses have shown a higher contrast in their overall appearance compared with the DNA‐unfilled. Two other examples are shown in Figure 16. The T4 phage image (Figure 16a) again is showing two versions on the DNA‐filled state (Hirokawa et al., 2004). A particular interest lies in the fiber structure spreading around the DNA‐ unfilled T4 phage. These may show DNA fibers escaped from the phage’s head. On the other hand, we can recognize a closed‐packed spiral structure inside another, which may correspond to DNA molecule. The rotavirus image in Figure 16b is showing also two features, but this is rather reflecting the diVerence in the coat protein shell structure (Taniguchi et al., 2004). The low‐contrast viral particles scattering around a high contrast one may correspond to the single‐shelled (double‐layered) version. On the other hand, the high contrast one may be the double‐shelled (triple‐layered) version. For the case of the double‐layered rotavirus, coat proteins are clearly visible. V. HILBERT DIFFERENTIAL CONTRAST TEM Hilbert diVerential contrast (HDC) displays nanostructures of thin specimen objects in a topographical manner (Danev et al., 2002; Nagayama et al., 2004a). The specific optical device to manipulate electron waves for HDC‐TEM is the half‐plane p‐phase plate, which appears to be quite distinguishable from the Zernike phase plate used in ZPC‐TEM, although both have to be placed on the back‐focal plane of the objective lens. This technique

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FIGURE 16. ZPC‐TEM images (300 kV) of ice‐embedded unstained T4 bacteriophages (from Hirokawa et al., 2004) and ice‐embedded unstained bovine group A rotaviruses (from Taniguchi et al., 2004). (a) An image of T4 phages. (b) An image of rotaviruses.

was once named diVerence contrast TEM (DTEM) but has renamed Hilbert diVerential contrast TEM after a literature reported on the application of the half‐plane p‐phase plate to LM (Lowenthal et al., 1967), where the authors named the method ‘‘optically processed Hilbert transform.’’ A. Contrast Transfer The half‐plane p‐phase plate works to change the phase by p for only those electrons penetrating through the plate (Figure 17a and b) (Danev et al., 2002). In the ideal situation where no loss of electrons occurs with phase plates, the wave function through the plate is modulated by a signum function, which takes 1 at the plate half‐plane and 1 at the open half‐plane (Figure 17c). As shown below, such a modulation can completely change the character of the CTF. For optically weak objects, a disturbance of waves immediately after the object (exit wave) is expressed as 1 þ aðrÞ þ ibðrÞðjaðrÞj; jbðrÞj 1Þ; here 1 represents the incidence without scattering, a(r) the object wave absorption (negative), and b(r) the change of object wave phase (negative). The lens‐ inducing phase retardation, g(k), is multiplied to the diVracted waves, dðkÞ þ ~ aðkÞ þ ~ bðkÞ in the form of eig(k). Real (cosg(k)) and imaginary (sin g(k)) components of the multiplier appear as modulation to the absorption a (r) and the phase retardance b(r), respectively. Placing a particular phase plate onto the back‐focal plane alters the CTF. In Table 5, CTFs characteristic to two TEM schemes so far discussed are compared with the one proposed in this section.

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FIGURE 17. Principle of Hilbert differential contrast TEM (from Figure 1 in Danev et al., 2002). (a) Schematics of HDC‐TEM with a half‐plane p‐phase plate set onto the back‐focal plane. (b) Top view of the half‐plane p‐phase plate. The direct beam (primary wave) passes through the open area close to the edge of the p‐phase plate. (c) Half‐plane p‐phase plate inducing signum‐function modulation development in the kx‐direction. (d) Sine‐type CTF developing in the kx‐direction, which is multiplied to the absorption term of waves. (e) Consine‐ type CTF developing in kx‐direction, which is multiplied to the phase term of waves.

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PHASE CONTRAST ENHANCEMENT TABLE 5 CONTRAST TRANSFER FUNCTIONS (CTFS) IN THREE CLASSES OF TEM (FROM TABLE 1 IN DANEV ET AL., 2002)

Phase CTF multiplied to b(k) Absorption CTF multiplied to a(k)

HDC‐TEM

DPC‐TEM

ZPC‐TEM

[2 cos g(k)] i sgn (kx) [2 sin g(k)]i sgn (kx)

2 sin g(k) 2 cos g(k)

2 cos g(k) 2 sin g(k)

Here, i represents the imaginary unit and sgn(kx) a signum function, which defines a function having a constant 1 in the positive domain and 1 in the negative domain along the kx axis (refer to Figure 17c).

Because the phase shift of waves is the major source of TEM imaging, the sin‐CTF instead of the cosine‐type one dominantly determines the image contrast. As shown in Table 5, aside from the term isgn(k), HDC‐TEM has a feature of the cos‐CTF completely equivalent to that of the ZPC‐TEM. Like ZPC, HDC‐TEM, therefore, could add high contrast to the obtained images. Therefore, the recoverage of the ZPC from the HDC is easily carried out by negating the image in the half‐plane of the k space, if the half‐plane plate does not show severe loss of electrons. But what is more significant with HDC‐TEM is the visual eVect of the diVerential contrast, which enables to visualize nanostructures in a topographical manner. The reason the half‐plane phase plate induces such an eVect can be discussed based on the point spread function of HDC‐TEM, which is obtained by Fourier transform of the corresponding CTF. To make things transparent, the odd‐natured phase‐CTF for HDC‐TEM (shown in Figure 17e) was approximated by a simpler function shown in Figure 18a (Danev et al., 2002). It is a combination of two square functions as, Pðkx =kc þ 12Þ þ Pðkx =kc  12Þ, where 8 1 1 > < 1;  < x < 2 2: PðxÞ ¼ > : 0; elsewhere The cutoV frequency kc corresponds to the first zero crossing in Figure 17e. The rapid sinusoidal modulation appearing in the frequency region higher than kc can be treated as bringing about an averaging out eVect to signals. The Fourier transform of the approximated function, namely the approximated point spread function of HDC‐TEM, is given by sin(pkcx) sinc(pkcx) (Figure 18b), where sincx ¼ sinx px . Figure 18b shows that the point spread function looks like a superposition of two d functions shifted to each

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FIGURE 18. The simplified CTF for HDC‐TEM (a) and the corresponding point spread function obtained by its Fourier transform (b) (from Figure 2 in Danev et al., 2002).

other by kc with opposite sign. The final image is given by the convolution of the spread function and the original wave function, which naturally leads to a diVerence contrast in the image and hence the topographic representation. Above consideration based on the approximation function must be corrected for the actual imaging, but the diVerence feature in the image could be retained. B. Comparison of Hilbert DiVerential Contrast and Defocus Phase Contrast TEM images of ultrathin sections of renal proximal tubular epithelial cells are shown in Figure 19 (Danev et al., 2002). With conventional electron microscopy (DPC‐TEM), the pictures are of low contrast due to high‐acceleration voltage and no staining, and hence, ultrastructures are

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not clearly observed (the upper row of Figure 19). With HDC‐TEM, on the other hand, ultrastructural details such as the cell body, cell organelles could be easily observed, as they are rimmed with the plasma membrane appearing as dark and bright lines (e.g., membranes as shown in Figure 19b, lower). In the cytoplasm, clusters of ribosomes are observed as obscure structures like clouds with DPC‐TEM (Figure 19a, upper), but each of ribosmal particles could be well visualized in the HDC‐TEM image (Figure 19a, lower). Ribosomes observed simply as electron‐dense particles in DPC‐TEM appear

FIGURE 19. Comparison of images obtained with HDC‐TEM and DPC‐TEM (300 kV cryo‐TEM; refer to Danev et al., 2002 on the experimental procedure and details of materials) (from Figure 3 in Danev et al., 2002). (a) Electron micrographs of a proximal tubular epithelial cell of rat kidney, osmium fixed, rapidly frozen, and freeze‐substituted tissue embedded in a resin. Ultrathin section observed with DPC‐TEM (upper) and HDC‐TEM (lower). Note that fine structures of various types of cell organelles were clearly visualized with HDC‐TEM for samples without conventional staining by heavy elements such as uranium and lead. (b) Electron micrographs of the same kind of specimen. They are observed with DPC‐TEM (upper) and HDC‐TEM (lower). Note the presence of granular structures in plasma membranes visualized with HDC‐TEM. Bar ¼ 0.2 mm.

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as electron‐dense particles rimmed with electron‐lucent areas in HDC‐TEM. This kind of rimming may enable the discrimination of each particle. In the nucleus, the nuclear envelope and its pores are not clearly visualized in DPC‐ TEM (Figure 19a, upper), but they could be well recognized by HDC‐TEM (Figure 19a, lower). The HDC‐TEM images shown in Figure 19 clearly illustrate the bulk topographic contrast. The images look similar to images produced by scanning electron microscope. However, the ‘‘height’’ in the HDC‐TEM images does not correspond to sample thickness variations (the sample was prepared with uniform thickness) but to the phase delay of the electron wave. This means that areas of higher electron density (heavier material) will be represented as being above their surroundings in the topographical image. If the sample does not have uniform thickness, the ‘‘height’’ in the topographical image will also correspond to the specimen thickness. The topographical image induced by the HDC may show diVerent appearance in image depending on the mutual direction between the target object and the half‐plane plate. The rotation of the half‐plane p phase plate induces the rotation in the direction the diVerential is taken, which finally results in an image appearance as if things are illuminated from diVerent angles according to the rotation. Historically, this kind of image representation was recognized in the images obtained with the knife‐edge aperture (refer to Section VI.A). Besides the diVerential representation, HDC‐TEM overall shows much higher contrast compared to DPC‐TEM (see Figure 19). This can be helpful for observing samples with low contrast, such as nonstained samples or samples observed with higher acceleration voltages (see Section V.D).

C. Significance of Lower Frequency Components HDC‐TEM is superior to ZPC‐TEM in the recovery of the lower spatial frequency components. Contrary to the fixed lower limit of the recoverable frequency with the Zernike phase plate, the half‐plane plate allows the adjustment of the limit in an arbitary manner, as the primary wave can be focused as closely as possible to the edge of the plate. A calculation assuming 50 nm for the distance between the primary wave focus and the plate edge for our 300 kV system resulted in the lower frequency limit to be around 0.004 nm1. The value should be compared with the lower frequency limit (0.04 nm1) realized for 300 kV ZPC images taken with a phase plate having a 1‐mm central hole. To observe the significance of the lower frequency limit for the contrast recovery, a model experiment had been made. The result is shown in Figure 20.

103 FIGURE 20. The origin of the high contrast in HDC‐TEM images. Original image (b) and corresponding diffractogram (a) of polystyrene latex particles. Lowest ky‐deleted diffractogram (c) and corresponding contrast‐dehanced image (d). Lowest k‐deleted diffractogram (e) and corresponding contrast‐dehanced image (f).

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Figure 20a shows an HDC‐TEM image taken for polystyrene latex beads with a geometry of about 60 nm in diameter. The quite highly contrasted image may reflect complete recovery of the low‐frequency component up to that corresponding to the bead diameter. Figure 20b is a simulated image based on the image of Figure 20a to investigate how the contrast is lessened when the focused primary wave is misaligned slightly far from the plate edge. Nulifying the lower frequency components along the x axis up to 0.08 nm1 induces quite a loss of the contrast as shown. By using the image of Figure 20a, we can also simulate to what extent the contrast is lost when the central hole of the Zernike phase plate is finite sized. Nulifying the frequency components around the origin to the extent corresponding to the 2 mm in diameter hole corresponding to the lower cut-oV about 0.08 nm1 decreases the contrast quite a lot, as shown in Figure 20c. This result indicates that the recovery of lower frequency components is significant for the imaging of relatively large objects. This is the major reason we have to rely on HDC‐TEM for the observation of biological specimens that are much larger than viruses. D. Biological Applications All of the experiments reported here were made with a 300 kV TEM system, as explained in Section IV.D. Experimental conditions were also the same as those described in that section, if not specified. 1. Cultured Cells Figure 21 shows an overview of HEK293 cells cultured on a TEM grid (Usuda et al., 2003), which was observed with a phase‐contrast light microscope (Olympus). Most of the cultured cells developing on the formvar film seem to be partially overlapping each other, but some of them are well isolated. One of the isolated, which is in the beginning of mitosis, is clearly visible in the center of Figure 21 and good for the TEM observation. HDC‐TEM images for the preparation are shown in Figures 22–24. Figure 22 is an image demonstrating a partial view of an HEK293 cell including various organelles (Usuda et al., 2003). The folded inner membrane structure developed inside the mitochondrion is closed in Figure 22b. Figure 23 is another cell image mainly demonstrating another kind of organelle, peroxizome (Usuda et al., 2003). A closeup view is shown in Figure 23b. The identification was done by the image appearance of clearly visible granules attributable to protein condensations. Figure 24 is the third example of organelle, endoplasmic reticulum (ER) (Usuda et al., 2003). A high closeup view is shown in Figure 24b. The

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FIGURE 21. Light microscopic image of unstained HEK293 cells as cultured over a formvar membrane on a TEM grid (from Usuda et al., 2003). The formvar membrane on the TEM grid was dipped in a Dulbecco’s MEM supplemented with 10% FBS.

FIGURE 22. HDC‐TEM image (300 kV) of an ice-embedded unstained HEK293 cell cultured over the formvar membrane as shown in Figure 21 (from Usuda et al., 2003). (a) Overview. (b) Closeup view for a mitochondrion‐looking organelle.

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FIGURE 23. HDC‐TEM image (300 kV) of an ice‐embedded unstained HEK293 cell as cultured over the formvar membrane as shown in Figure 21 (from Usuda et al., 2003). (a) Overview. (b) Closeup view for a peroxizome‐looking organelle.

FIGURE 24. HDC‐TEM image (300 kV) of an ice‐embedded unstained HEK293 cell as cultured over the formvar membrane as shown in Figure 21 (from Usuda et al., 2003). (a) Overview. (b) Closeup view for an endoplasmic reticulum–looking organelle.

identification for ER is much poorly evidenced compared with mitochondria and peroxizomes, but granular particles scattering on the membrane resemble ribosomes, which characterize ER. These examples show the observability of biological specimens without staining once TEM is properly phase plated, as shown here. Our expectation on the TEM observation of cultured whole cells was that they must be too thick to work even with an acceleration voltage of 300 kV.

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Therefore, photo images shown in Figures 22–24 were really sort of a surprise to us. Actually, the thickness limit for the conventional TEM has been thought to be around 200 nm and even for the tomography around 500 nm. The thickness of HEK293 cells varies depending on cell areas and may be in the range 1–10 mm. Therefore, it was expected that any area in the cell could not fulfill the weak object condition. Now we are tentatively interpreting the astonishing results due to a lucky combination of the energy‐filtering eVect and the eventual satisfaction of the observed area for the object‐dependent phase retardance y to be within np < y < np þ p/2. This significant finding is discussed in Section VIII in the context of the weak object condition. 2. Cyanobacterial Cells Compared with cultured cells, bacteria belong to an easy sample because they are much smaller and thinner. An example is shown for the case of cyanobacteria (Synechochoccus sp. PCC7942), which is a kind of bacteria including chloroplasts and has a cylindrical geometry of 3 mm (length)  1 mm (diameter). Figure 25 shows a comparison of images obtained for an unstained ice‐embedded whole cell and a stained sectioned cell (Kaneko et al., 2005). The diVerential feature, together with the high contrast, is demonstrated in Figure 25a for an ice‐embedded whole cell. Counterexamples obtained with DPC‐TEM are shown in Figure 25b and c. An obscure structureless image is observed in the DPC for the sample shown in Figure 25b, which was taken under the same experimental conditions as for Figure 25a, except for the phase plate and the defocus. The unexpectedly large diVerence in contrast between the two images (Figure 25a and b) is likely attributable to the diVerence in the CTF, coupled with the large defocus variation happening in a thick sample. The cosine CTF characterizing HDC‐TEM is less sensitive to the variation of defocus than the sine CTF once the just focus is set near the depth center in the sample. The defocus needed to make the first zero of the cosine CTF coincide with the Nyquist frequency was calculated to be 3.2 mm. Because no cyanobacterial structure was discerned with DPC‐TEM with a slight defocus close to the just focus used for the HDC‐TEM (data not shown), the DPC‐TEM image was taken at a deep defocus of 15 mm to obtain an image with the best contrast as an illustration of what is attainable by DPC‐TEM. Comparing another pair of images, the ice‐embedded whole cell (Figure 25a) and the resin‐embedded sectioned cell (Figure 25c), we recognize a large diVerence in the image appearance, which may be attributable to the

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FIGURE 25. Comparison of TEM images of cyanobacterial cells (from Figure 1 in Kaneko et al., 2005). (a) 300 kV HDC‐TEM image of an ice‐embedded unstained whole cell (near‐focus). (b) 300 kV DPC‐TEM image of the same ice‐embedded unstained whole cell as shown in (a) (
15 mm defocus). (c) 100 kV DPC‐TEM image of a resin‐embedded, sectioned, and stained cell.

enormous diVerence in specimen treatment. In the sectioned cell, first of all, we see a ragged cell wall, which indicates that some shrinkage of the cell has occurred during the TEM preparation. Many aggregates and associated voids are also recognized, which are inevitably induced by chemical treatment, such as dehydration and selective staining of cellular organelles. On the other hand, the images of the ice‐embedded cell are smoothly round and recognizably space filled everywhere. Notice here that the rapid freezing is expected to preserve the overall structure, such as the cell shape, as well as subcellular structures. The preserved roundness of cyanobacterial cells allows us to estimate the specimen thickness to be about 1 mm. Considering the deep focal depth of the 300‐kV HDC‐TEM, the image obtained should be a projection image of the 1 mm thick specimen. We did

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not expect that individual organelles, proteins, and DNA could not be visualized in the image with such a thick projection, because the partial structures should overlap. However, as we present here, the ultrastructural details could be recognized in this condition. This fact was the second surprise to us. There are two possible explanations for this unexpected result. First, cyanobacteria have relatively sparse structures and abundant water content. Second, in the case of the thylakoid, high contrast was gained because of its massive structure and overlapping of constituent lipid molecules in the direction of focal depth. Obviously none of these images could be obtained without HDC‐TEM. Novel features of intact subcellular structures found in cyanobacterial cells are shown in Figure 26 (Kaneko et al., 2005). Closeup views of inner portions of cells and putative ultrastructural identifications are shown. The four examples provided were gathered from various cell images based on their structural similarities to DPC‐TEM images taken for resin‐embedded cells. Past reports using DPC‐TEM were referred to for the identification of thylakoid, Rubisco, and phycobilisome. Although the structural similarities and distribution within the cells were used for the identification, the procedure was not a simple pattern recognition. As is characteristic of DIC microscopy, HDC adds completely novel morphological features to TEM images. This has made the comparison between the conventional thin‐section image (Figure 25c) and the ice‐embedded one (Figure 25a) an unconventional task. Nevertheless, there are several ultrastructural features that can be more clearly recognized by HDC‐TEM than by conventional methods. One can see numerous arrays of globular structures (circled in Figure 26a) that are embedded in thylakoid (Sherman et al., 1994) membranes, possibly components of photosystems or other membrane proteins. Some of these structures have an elongated shape and appear to be sticking out of the membrane. Many polyhederal bodies (carboxysomes) can be recognized easily, and, moreover, the constituent Rubisco (Oru´ s et al., 1995) molecules (Fogg et al., 1973) (circled in Figure 26b) can be distinguished. Phycobilisomes (circled in Figure 26c) appear to be attached to thylakoid membranes through rod-like structures (arrows in Figure 26c). These detailed ultrastructures revealed by HDC‐TEM must be associated with specific functions, which have yet to be determined. 3. Isolated Organelles Two examples are shown on the isolated organelles obtained from mammalian cells. Images for the isolated microtubles are shown in Figure 27 (Usuda et al., 2003). The comparison of the DPC and the HDC for the same sample

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FIGURE 26. Closeup views of various structures from cyanobacterial cell HDC images (from Figure 2 in Kaneko et al., 2005). Scale bars, 100 nm. (a) Thylakoid‐like ultrastructural images. (b) Rubisco‐like ultrastructural images. (c) Phycobilisome‐like ultrastructural images.

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FIGURE 27. 300 kV cryo‐TEM images of so‐called ‘‘twice‐cycled’’ microtubules obtained from the supernatant of a rat brain homogenate in the presence of GTP (from Usuda et al., 2003). (a) DPC‐TEM image. (b) HDC‐TEM image taken for the same area as shown in (a).

under the same condition except for the phase plate is again clarifying the contrast diVerence between the two contrast schemes, the sin‐CTF (Figure 27a) and the cos‐CTF (Figure 27b). The diameter of microtubles, about 10 nm, is relatively small for the HDC to be eVective. Nevertheless, the HDC image detailing inner structures is far better than the conventional. To have better contrast and higher resolution for microtubules, the thickness of ice has to be carefully adjusted to be thin comparable to the microtubule thickness. The second example is the ice‐embedded mitochondria, which was prepared from a mutated human cell line, a mitochondria‐associated disease model, in which mitochondria are starved by a metabolic deficiency. The HDC‐TEM for the genetically manipulated mitochondria is shown in Figure 28 (Matsumoto et al., 2004). The mitochondrial size, about 1 mm, is problematic for the clear image when they are densely filled with constituent protein molecules. Normal mitochondria in this sense is too dense to be clearly observed even with HDC‐TEM. The starved mitochondria, on the other hand, has resulted in clear images, as shown in Figure 28. We can recognize various membrane structures (Figure 28a) and granular structures (Figure 28b) in the disease model mitochondria, which have to be further studied in terms of pathology.

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FIGURE 28. HDC‐TEM images (300 kV) of ice-embedded unstained mitochondria isolated by sequential differential centrifugation of homogenates from the cybrid cell line 2SD with a mutation in the t‐RNA gene, which deletes in electron signaling and ATPase systems (a and b) (from Matsumoto et al., 2004).

VI. FOUCAULT DIFFERENTIAL CONTRAST TEM Phase contrast methods so far discussed with or without phase plates are applicable only to weak objects that perturb the incidence with a phase retardance smaller than p/2. In this section, a novel phase retrieval technique applicable to strong objects is introduced. The innovation core is the dynamic use of a knife edge, which is usually fixed (Nagayama, 2004). The synchronous operation between the scanning of the knife edge and the image accumulation enables a novel spatial filter, which draws the object‐ dependent phase retardance in the form of its first derivative. A. Contrast Transfer 1. Foucault Knife‐Edge Scanning Filters The essence in converting object‐dependent phase retardance to the phase contrast in Schlieren optics lies in the asymmetric masking with a Foucault knife edge of the Fourier space (k‐space) as shown in Figure 29a. Half‐plane masking is common in this method, but any degree of masking with a knife edge, whether it masks over or under 50% coverage, must induce a phase contrast to some extent. Then as for the knife‐edge function, we can pose questions such as ‘‘How does the phase contrast vary according to the degree of masking of the knife

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FIGURE 29. Schlieren optics with a Foucault knife‐edge (from Figure 1 in Nagayama, 2004). (a) Knife‐edge halfly masking the back‐focal plane of an objective lens. (b) Image formation accompanying a spatial filter, H(kx), represented by Fourier transform (FT ) ~ x ; ky Þ ¼ FT½Cðx; yÞ; Cðx; yÞ ¼ FT 1 processes. The lens magnification is set to one. Cðk ~ ½Cðkx ; ky Þ. HðxÞ ¼ FT 1 ½Hðkx Þ. FT [. . .] and FT1[. . .] indicate FT and inverse FT.

edge?’’ and ‘‘What happens when many images are accumulated during scanning of the knife edge?’’ An innovative spatial filter is the answer, which can be called a Foucault knife‐edge scanning filter or, in short, Foucault diVerential filter. Figure 30a and b schematically depicts how to shoot images with the knife‐edge scanning. The left knife‐edge scanning image (IL(r)) is taken by moving the knife edge from the right to the left or the left to the right as shown in Figure 30a. The right knife‐edge scanning image (IR(r)) is taken as shown in Figure 30b. Here, left and right are used in an average sense. For example, looking at the back‐focal plane in the light‐going direction, the left‐scanning filter shades the left side of the plane on average. 2. Theory of Biased Derivative Filters A derivative filter is a pure intensity filter acting in the k‐space and gives a derivative of the wavefront for images in the real space given as

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FIGURE 30. Phase retrieval experiments using knife‐edge scanning filters (from Figure 2 in Nagayama, 2004). The knife‐edge movement is schematically shown as (a) Left‐knife‐edge scanning filter when the left side of the back‐focal plane is shaded [shooting IL(r)]. (b) Right‐ knife‐edge scanning filter when the right side of the back‐focal plane is shaded [shooting IR(r)]. Left or right is defined when the back‐focal plane is observed from the incidence-going direction. Opposite movements of the knife edge are possible as shown in (a) and (b), and the order from open to close or vice versa does not matter.

Derivative filter : Hd ðkÞ ¼ kx ða linear function in the Fourier spaceÞ: ð26Þ With an appropriate lens system where image formation is a process of a Fourier transform (FT) and a successive inverse Fourier transform (FT1) for wavefronts (C(r)) (as shown in Figure 29b), the function of the derivative filter is expressed as 1 1 d FT filter ~ ðkÞ  ~ ðkÞ FT C ðrÞ: ! kx C ! CðrÞ ! C i2p dx

ð27Þ

For simplification, the lens magnification is ignored. We propose novel filters that are an extension of the derivative filter, 1 Biased left-derivative filter : BL ðkx Þ ¼ ð1 þ kx Þ; ð28Þ 2 1 Biased right-derivative filter : BR ðkx Þ ¼ ð1  kx Þ: 2

ð29Þ

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The function of these filters becomes visible when they are applied to a wavefront (WF) C(r) ¼ eiy(r). The observation of objects through these biased derivative filters is expressed as 1 ½B ðk ÞFT½CðrÞ Biased left-derivative filtered WF : DL ðrÞ ¼ FT L x   1 1 dCðrÞ ¼ CðrÞ þ 2  i2p dx  1 iyðrÞ 1 dyðrÞ ¼ e 1þ ; 2 2p dx

ð30Þ

BR ðxÞFT½CðrÞ Biased right-derivative filtered WF : DR ðrÞ ¼ FT 1 ½  1 iyðrÞ 1 dyðrÞ ð31Þ ¼ e 1 : 2 2p dx The actual images include the intensity of these WF. Biased left‐derivative filtered image: "   # 1 1 dyðrÞ 1 dy 2 2 1þ þ 2 IL ðrÞ ¼ jDL ðrÞj ¼ : 4 p dx 4p dx Biased right‐derivative filtered image: "   # 1 1 dyðrÞ 1 dy 2 2 1 þ 2 : IR ðrÞ ¼ jDR ðrÞj ¼ 4 p dx 4p dx

ð32Þ

ð33Þ

3. Knife‐Edge Scanning Filters that Realize Biased Derivative Filters Biased derivative filters shown above are only theoretical, and intensity filters for real objects must always be associated with phase retardances, regardless of the modulation source (e.g., electric field modulation, magnetic field modulation, or thin‐film modulation). Pure intensity filters can become real only when the filters are dynamically manipulated, as shown next. The knife‐edge function is to divide the back‐focal plane into two areas. In one area, the incidence is completely intercepted, and in the other, the incidence is completely transferred. Then, the filter function is mathematically expressed as    1 j 1  sgn kx  ; j ¼ N; ...; N; ð34Þ 2 N where sgn kx is the signum function (sgn(x) ¼ 1(x  0), ¼ 1(x < 0) ) and j/N is the position of the knife edge and represents the boundary between the

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completely transferred and the completely intercepted area. Namely, j goes from N to N, when the moving range (say (1, 1)) is divided by 2N equidistances. The positive sign corresponds to the knife edge shading the left side, and the negative sign corresponds to the knife edge shading the right side. With Eq. (34), the function of the left or right knife‐edge scanning filter is expressed as   N   Ð 1 X j 1 1 lim 1  sgn kx  ¼ 1  dkx Uðkx Þ ¼ ð1  kx Þ; ð35Þ N!1 4N N 2 2 j¼N where U(kx) is the square function (U(kx) ¼ 1 (|kx|  1), 0 (|kx| > 1)). The sign  corresponds to Eq. (34). Equation (35) indicates that the left knife‐edge scanning filter 1/2 (1 þ kx) is realized when the knife edge is moved to the left from the completely closed state ( j ¼ N) to the completely open state ( j ¼ N), which is shown in Figure 30a. On the other hand, the right knife‐ edge scanning filter is obtained when the knife edge is moved to the right from the completely closed to the completely open, as shown in Figure 30b. The premise for the left or right knife‐edge scanning filter to function exactly as described by Eq. (35) is the superposition rule applied to the bilinear interference component that is involved in the squarely detected images. This is shown below. The filter function of the knife edge when centered on the back‐focal plane is expressed as knife‐edge half‐plane filter:   
1 1 iyðrÞ FT iyðkÞ filter 1 iyðkÞ FT 1 iyðrÞ iyðrÞ e e e ! ~e ! !  1  sgnðkx Þ ~e : ð36Þ 2 2 ipx Left or right knife‐edge‐filtered image: 2 !      1 1 1 1  Re eiyðrÞ  eiyðrÞ þ   eiyðrÞ  IL ðrÞ ¼ 4 ipx ipx 2 !     1  1 iyðrÞ 1 iyðrÞ iyðrÞ  1 þ Re e e e IR ðrÞ ¼ þ   4 ipx ipx

ð37Þ

ð38Þ

The interference term RebeiyðrÞ 1=ipx  CðrÞc (asterisk represents convolution) involved in the above knife‐edge–filtered images is bilinear to eiyðrÞ and 1/ipx*eiyðrÞ , and only the term 1/ipx*eiyðrÞ varies according to the shift of the knife‐edge position to j/N. Therefore, the linear superposition represented by Eq. (35) can also survive the intensity detection, which leads to the linearly filtered images. Finally, this is the function represented by the

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biased left‐ or right‐derivative filter shown in Eqs. (28) and (29), and the biased derivative–filtered images expressed by Eqs. (32) and (33) are obtained. In Eqs. (37) and (38), two square terms (the first and third) are observed in addition to the bilinear term. These undesired terms can be cancelled in the diVerence image I (r) because they contribute equally to the left and right knife‐edge scanning filter. On the other hand, in the summation image Iþ(r), the bilinear term is cancelled, and the first term, which corresponds to the conventional image (C(r)), remains. The third term represents a deviation from C(r), which can be neglected when the deviation is small. When the object is absorptive, the wavefront C(r) changes to A(r)eiy(r). For such a mixed object, Eqs. (32) and (33) are converted to more general forms as ( "   2 #) 1 2 1 dyðrÞ 1 dA 2 dy ð39Þ þ 2 þ IL ðrÞ ¼ A ðrÞ 1 þ 4 p dx 4p dx dx ( "    2 #) 1 2 1 dyðrÞ 1 dA 2 dy þ 2 IR ðrÞ ¼ A ðrÞ 1  þ : 4 p dx 4p dx dx

ð40Þ

To obtain pure images corresponding to the phase derivative, we can employ the diVerence of the two biased filtered images as I ðrÞ ¼ IL ðrÞ  IR ðrÞ ¼

1 2 dyðrÞ A ðrÞ : 2p ap

ð41Þ

By combining C(r) or IL(r) þ IR(r), finally, we obtain a pure phase derivative image: IF ðrÞ ¼ I ðrÞ=CðrÞ ¼

1 dyðrÞ ffi I ðrÞ=½IL ðrÞ þ IR ðrÞ 2p dp

ð42Þ

We term the phase contrast that can be obtained using Foucault knife‐ edge scanning as the Foucault diVerential contrast. For example, IL(r) or IR(r) is termed the Foucault left or right diVerential contrast image, I–(r) as the Foucault diVerential diVerence contrast image, and IF (r) as the Foucault pure diVerential contrast image. B. Numerical Simulations This innovation is applicable to any imaging system that uses lenses. We demonstrate this using simulated images.

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Computer simulations for knife‐edge scanning-filtered TEM images were performed for a conically shaped model object, which was modeled by combining a pure phase object (bright cone in Figure 31) and a purely absorptive object (dark cone in Figure 31). Two kinds of TEM observations, which correspond to the left knife‐edge scanning filter (IL(r)) and the right knife‐edge scanning filter (IR(r)), were separately simulated for the object. The maximum absorption at the cone summit of the pure absorption object is 0.5, as determined by the transparency index. The maximum phase retardation at the cone summit of the pure phase object is 2p. The knife edge ranges between 0.1 nm1 and 0.1 nm1. Figure 31 shows a bird’s eye view (c), a top view (d), and a side view (e) of the model. Figure 32a shows a top view and summit‐through cross‐sections (A and B) for the two cones of the left knife‐edge scanning-filtered image. The A cross‐section corresponds to that of pure absorption object, and the B cross‐section corresponds to pure phase object. Although the filter typically intercepts the left side of the back‐focal plane, the phase derivative image of the pure phase cone shows characteristic shading in the right side. The profile of the A cross section for the absorptive object is a concave triangle rimmed by a linear function (x) due topthe ffiffiffi square detection of inverse‐parabolically shaped absorption object ð xÞ. On the other hand, the profile of the B cross-section for the phase object is a linear function with a positive definite in the left side and a negative definite in the right side as expected for the derivative of a phase object with a parabolic shape [(a þ x)2 for a < x < 0 in the left side and (a  x)2 for 0 < x < a in the right side]. Anomalies are observed at the summit and boundaries, which may be

FIGURE 31. Model object used for the image simulation (from Figure 3 in Nagayama, 2004). The object is a combination of a pure absorption object and a pure phase object. The vertically cross‐sectional shape for the pure phase object (bright cone) is a vertical parabola (x2) pffiffiffi and that for the pure absorption object (dark cone) is a horizontal parabola ð xÞ.

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attributed to the abrupt shape change, and the high‐frequency components above 0.1 nm1 are cut with the square aperture. Figure 32a shows a top view and cross‐sections of the right knife‐edge scanning filter image. When compared to Figure 32b, the shape of the A cross‐section is equivalent, but as expected, the B cross‐section shows an inverse slope, which corresponds to the left-side shading on average. With the complementary knife‐edge scanning images, a pure phase derivative image free from amplitudes can be obtained, as shown in Figure 32c. To illustrate the remarkable characteristics of knife‐edge scanning filters, a simulated image obtained with the conventional Foucault knife‐edge (Schlieren) method for the conical model is shown in Figure 33. As expected from Eqs. (37) and (38), with an explicit bilinear term, the Schlieren images, especially for a pure phase object, have severe modulation with a long tail, which is due to the convolution between 1/x and the original wavefront C(r). The knife‐edge scanning filter can trim the tail, as shown in Figure 32.

VII. COMPLEX OBSERVATION in TEM In Section II, the complementarity of two contrast schemes, the DPC and the ZPC, has been deliberately emphasized. The complex observation discussed in this section uses the complementarity in an analytical fashion. This observation scheme is based completely on the coherent microscopy and is composed of two or three experiments consisting of twin experiments restoring two linear terms corresponding to the real and imaginary part of complex images (Nagayama, 1999). A linear combination of two images obtained through DPC‐ and ZPC‐TEM is able to exhibit a complex quantity in the form to be numerically manipulated without rupture of the formal theory of image formation. The basic scheme is applied to settle the long‐ standing issue in electron microscopy that the image is deteriorated by the modulation during the contrast transfer. The full description of the CTF theory is given.

A. Basic Scheme and CTF Demodulation First, an idealized complex observation, where idealized filters and no CTF modulation are assumed, is developed. According to the formal theory of image formation, the optical signal C(r) arriving at the image plane, which is complex and carried by a carrier wave C0, is expressed as

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FIGURE 32. Simulated images with knife‐edge scanning filters (from Figure 4 in Nagayama, 2004). (a) Left knife‐edge scanning image [IL(r)]. A top view (left) and

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FIGURE 33. A simulated image with a Foucault knife‐edge, which intercepts electrons at the left side of the back‐focal plane (Schlieren method) (from Figure 5 in Nagayama, 2004). A top view and cross‐sections as explained in Figure 32a.

CðrÞ ¼ C0 ZðrÞ;

ð43Þ

ZðrÞ ¼ aðrÞ þ ibðrÞ:

ð44Þ

Here Z(r) represents a 2D complex image at the image plane system r, a(r) and b(r) correspond to the real and imaginary part of the image, respectively, and a(r) comprehensively includes system characters irrespective of the image, such as the illumination strength and image formation conditions. C0 is basically a complex quantity representing the carrier wave, but it can finally be converted to a real constant and safely ignored. Now we introduce a condition important to our complex observation (Nagayama, 1999): ð1 ð1 C ðrÞd r ¼ Z ðrÞd r 6¼ 0 ð45Þ S0 ¼ 1

1

In any optical system treated at least in laboratories, the above condition can readily be satisfied by adjusting the area covered by objects. S0 corresponds to the 0‐th order diVraction. Without losing generality, the signal cross‐sections for the pure absorption object (A) and the pure phase object (B). (b) Right knife‐ edge scanning image [IR (r)]. A top view and cross‐sections as explained in (a). (c) Pure phase image in the derivative form obtained by the difference of IL(r) and IR(r) divided by the sum of IL(r) and IR(r). A top view and cross‐sections as explained in (a).

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expression can be normalized by this integral. We have a new definition: ð1 ZðrÞdr ¼ 1 ð46Þ CðrÞ ¼ ZðrÞ; 1

On this ground, a combination of three experiments mentioned below completes the complex observation, which can convert the fractional information held in the respective images into a unified one, faithful to the original complex signal. Exp I: Conventional microscopy (DPC) to extract the real component from the complex WF (see Figure 4a). The original optical image and the detected signal are given by CI ðrÞ ¼ ZðrÞ ¼ 1 þ DðrÞ; DðrÞ ¼ ZðrÞ  1;

ð1 1

DðrÞdr ¼ 0;

I I ðrÞ ¼ CI ðrÞCI ðrÞ ¼ 1 þ 2Re½DðrÞ þ jDðrÞj2 ;

ð47Þ ð48Þ ð49Þ

where D (r) represents a background‐free image and the unity, 1, in Eq. (47) stands for the primary wave without diVraction (scattering), which contributes to the background. This background appears as the 0‐th order diVraction at the back‐focal plane. Exp II: Phase‐contrast microscopy (ZPC) to extract the imaginary component (see Figure 4b). The primary wave corresponding to the background is shifted from 1 to i by inserting a p/2 phase plate at the center of the back‐focal plane. The complementary optical signal is expressed as CII ðrÞ ¼ 1  iDðrÞ;

ð50Þ

I II ðrÞ ¼ CII ðrÞCII ðrÞ ¼ 1 þ 2Im½DðrÞ þ jDðrÞj2 :

ð51Þ

Exp III: Dark‐field microscopy to reproduce the square term of the complex image. The primary wave contributing to bright field is completely intercepted by a stopping microplate inserted at the center of the back‐focal plane. This signal is given by CIII ðrÞ ¼ DðrÞ;

ð52Þ

I III ðrÞ ¼ CIII ðrÞCIII ðr Þ ¼ jDðrÞj2 :

ð53Þ

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The combination of Eqs. (49), (51), and (53), by using a complex summation, which is symbolically given by [Exp I]þi[Exp II] (1 þ i)[Exp III], finally retrieves the original image into its complex form as I t ðrÞ ¼ I I ðrÞ þ iI II ðrÞ  ð1 þ iÞI III ðrÞ ¼ ð1 þ iÞ þ 2DðrÞ ¼ ði  1Þ þ 2ZðrÞ:

ð54Þ

The first term or the right‐hand side is a constant and must numerically be manipulated by the Fourier transform. The three‐experiment scheme was developed to include the application to strong objects. If the weak object condition vital in the bright‐field optics, |D(r)| 1, is assumed, the triple experiment can be replaced by the double one by ignoring the dark‐field mode. The proposed scheme is applied to electron microscopy to overcome the issue of the CTF demodulation. The manipulation of the primary wave is carried out only at the back‐focal plane by keeping other device and sample conditions unchanged. In the actual image formation, object images are generally to be blurred by the modulation due to the CTF as discussed in previous sections. The CTF is resulted from the lens‐dependent phase retardance g(r). This process is formulated as
 CF ðkÞ ¼ SF ðkÞAðkÞexp igðkÞ ; ð55Þ SF ðkÞ ¼ FT½SðrÞ

ð56Þ

SðrÞ ¼ C0 zðrÞ;

ð57Þ

zðrÞ ¼ aðrÞ þ ibðrÞ;

ð58Þ

where CF (k) represents the FT image (diVraction) of an object optically blurred by lens aberrations, A(k) represents the aperture which determines the range of spatial frequency contributing to the image formation. Exp(ig (k)) corresponds to the CTF as explained. S(r) is an optical signal diVracted from the object including the primary wave, C0 represents the plane, wave of incident electrons. z(r) represents the complex amplitude defined in the exit plane soon after the object. In this formulation, the two coordinate systems in the object and image planes are simply denoted by the same symbol, r. We also neglect the change of scale induced by the lens magnification. Real three‐dimensional objects here are theoretically treated as 2D ones. This means that optical information in the z‐direction is projected to the two

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FIGURE 34. Lens‐dependent phase retardance and the associated contrast transfer functions induced by the spherical aberration and the defocus (from Figure 3 in Nagayama,

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2D functions, a(r) and b(r). As known well, there is a limitation on the object thickness for the assumption to hold. The lens‐dependent phase retardance g(k) is given in Eq. (20), but here ^ and normalized defocus duplicated by using the normalized wave number, k D^z,   1 ^4 D^z2 ^ gðkÞ ¼ 2p k  ; ð59Þ 4 2 ^ ¼ ðCs l3 Þ1=4 jkj; D^z ¼ ðCs lÞ1=2 Dz: k

ð60Þ

Due to the rapid growth of g(k), as shown in Figure 34a, CTFs defined by ^ (Figure 34b sin g(k) or cos g(k) oscillates very rapidly for higher values of k and c) and hence severely modulates the Fourier image SF (k). This is the major cause of the blurred image and the lowered contrast in electron microscopy. The phase objects are invisible, but as discussed in Section I.B, the invisibility can be recovered by controlling g(k) through defocus. Let us precisely formulate the contrast mechanism by following Scherzer’s theory. Neglecting the factor C0, the optical signal representing objects is simply written as ð1 zðkÞdr ¼ s0 : ð61Þ SðrÞ ¼ zðrÞ ¼ s0 þ DðrÞ; 1

The constant term s0 in the right‐hand side of Eq. (61) corresponds to the background arising from the primary wave. The term D(r) is a background‐ free object function. The function SF (k) at the back‐focal plane is given as SF ðkÞ ¼ s0 dðkÞ þ FT½DðrÞ:

ð62Þ

Putting Eq. (62) into Eq. (55), and then inverse Fourier transforming it, we obtain the object image for the Exp.I.
 CI ðrÞ ¼ FT 1 ½fs0 dðkÞ þ FT½DðrÞgAðkÞexp igðkÞ  ð63Þ ¼ s0 þ DðrÞ;

1999). (a) Lens‐dependent phase retardance plotted along the normalized wave number, k^. (b) Since type of contrast transfer function. (c) Cosine type of contrast transfer function. The number pattached to each of the functions is the defocus value, Dz^. The particular value ffiffiffiffiffiffi D^z ¼ 3= 2p ¼ 1:197 corresponds to the optimum defocusing proposed by Scherzer (the Scherzer focus).

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 DðrÞ ¼ DðrÞ  FT 1 ½AðkÞexp igðkÞ 
 ¼ zðrÞ  FT 1 ½AðkÞexp igðkÞ :

ð64Þ

For the derivation of Eq. (64), equalities such as exp(ig(0)) ¼ 1, A(0) ¼ 1, d(0)A(0) ¼ 0, FT 1[1] ¼ d(k), and FT 1[d(k)] ¼ 1 are used. For the normalized signal, by replacing s0 by 1, the square detection finally leads to I I ðrÞ ¼ jCI ðrÞj2
 ¼ 1 þ 2RefzðrÞ  FT 1 ½AðkÞexp igðrÞ g þ jDðrÞj2 ¼ 1 þ 2aðrÞ  FT 1 ½AðkÞcosgðkÞ  2bðrÞ  FT 1 ½AðkÞsingðkÞ þ jDðrÞj2 :

ð65Þ Comparing Eq. (64) with Eq. (49), we find the image deformed by the factor 2FT 1 ½AðkÞexpðigðkÞÞ. In the actual computation, the constant term in Eq. (65), which is complex in general, becomes indefinite due to the equality d(0)A(0) ¼ 0. The background, therefore, must be separately adjusted to be faithful to that of the object function in the final stage. As shown in Figure 34b, the sin‐CTF becomes weak and even zero near ^ (Nagayama, 1999). It has also many zeros along the k ^ axis, the origin of k which gives rise to the lethal problem in electronpmicroscopy, as has been ffiffiffiffiffiffi discussed. A specific CTF with the D^z value of 3= 2p, corresponding to the Scherzer focus, which is thought to be a compromise between the spherical ^ axis aberration and defocus, is given in Figure 34b. This function cuts the k around 1.4, as shown in Figure 34b, which defines the Scherzer limit. As discussed in A, the way to get rid of the severe CTF modulation is to retrieve image signals in their complex form. The multiplication by sin(g(k)) on cos(g(k)), which takes many zeros, is not invertible, but the exp(ig(k)) multiplication is completely free from that sort of problem. After the combination of two complementary components with the complex summation as expressed by Eq. (54), we have
 ð66Þ I t ðrÞ ¼ i  1 þ 2zðrÞ  FT 1 ½A0 ðkÞexp igðkÞ ;
 ð67Þ ZðrÞ ¼ zðrÞ  FT 1 ½A0 ðkÞexp igðkÞ ; where the square component is ignored. This is exactly the form the reconstructed complex images should keep for successful CTF demodulation. The multiplication of the reciprocal of exp(ig(k)), called the inverse filter, can correct the modulated image as

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FTbI I ðrÞc  ½expðigðkÞÞ ¼ ði  1ÞexpðigðkÞÞdðkÞ þ 2FT½zðrÞAðkÞ; ð68Þ FT 1 fFTbI t ðrÞc  ½expðigðkÞÞg ¼ i  1 þ 2zðrÞ  FT 1 ½AðkÞ:

ð69Þ

When Eq. (69) is compared with Eq. (54), the corrected object image is found to be ZðrÞ ¼ zðrÞ  FT 1 ½AðkÞ:

ð70Þ

The aperture function A(k) restricts the microscopic resolution and modulates the image. In contrast to conventional electron microscopy where the Scherzer limit is thought to set the upper limit, the aperture limit can be extended two or more times wider because of the elimination of the CTF modulation. Comparing also with electron holography, the complex electron microscopy is advantageous in using whole frequency range in the k space, which ensures the three times improvement in resolution, if the other experi^ value is mental conditions are set equal between the two. The extension of k in principle limited only by the resolution of the image recording. Lastly, the experimental procedure of the two‐experimental version is summarized in Figure 35.

FIGURE 35. Flowchart of the computational procedure to reconstruct a CTF‐demodulated image from a pair of DPC‐ and ZPC‐TEM images.

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FIGURE 36. Complex reconstruction scheme applied to a pair of images obtained with a TEM working at 400 kV for a negatively stained ferritin sample (from Figure 2 in Danev et al., 2001a). Dofocus ¼ 765 nm. Scale bars represent 10 nm in (a) to (d), and 1nm1 in (e) to (g). (a) An image obtained with DPC‐TEM for a sample area. (b) An image obtained with ZPC

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B. Experimental Verification The scheme developed for the CTF demodulation can be applied to Zernike phase contrast and Hilbert diVerential contrast TEM in the combination of the conventional observation. As has been discussed in the previous sections, the two phase contrast schemes, ZPC and HDC, bear images modulated with the cos‐CTF. In the case of weak or medium objects complex images obtained with the combination of the conventional and the phase contrast can suYce the complex observation as a two experimental scheme. One of the examples obtained for the combination of the conventional (DPC) and the Zernike is shown in Figure 36 (Danev et al., 2001a). Figure 36 is self‐explanatorily illustrating the ZPC complex observation scheme. Particular interest in the very low contrast observed for the DPC image (Figure 36a) even for the negatively stained protein molecules, which must be compared with the high‐contrast ZPC image (Figure 36b). This is due to the high acceleration voltage used (400 kV) and the near‐focus condition. Nevertheless, it is useful to recover the pure phase and pure amplitude image shown in Figure 36c and d, where qualified images of ferritin molecules free from the CTF modulation are seen compared with the originals (Figure 36a and b). An example obtained for the combination of the conventional and the Hilbert is shown in Figure 37 (Danev et al., 2004). Figure 37 is also self‐explanatory on the HDC‐complex observation scheme, where nonbiological specimen, graphite, is used as an image example. Figure 37a and b show the image areas selected for complex reconstruction. Before searching for the same image areas the HDC‐TEM image was demodulated to recover the symmetric point spread function from the antisymmetric one, as shown in Figure 18b,c by negating the Fourier image in the negative half‐plane (Figure 37b). Without image demodulation, there

image for the same sample area shown in (a). (c) A CTF‐demodulated phase image reconstructed from the two complementary images shown in (a) and (b). This complementary pair displayed a phase shift of (0.38  0.03) p. (d) A CTF‐demodulated amplitude image paired to that shown in (c). (e) A sin g (k) modulation. The two black spots symmetrically spread about the origin represent the pinhole projection. (f) A cos g(k)‐modulated Fourier image showing the cosine‐modulated Thon ring. The image contrast around the origin is fairly preserved due to the nature of the cosine modulation. (g) A CTF‐demodulated Fourier image obtained from the two images shown in (e) and (f). Fourier transform (FT) is in principle reversible, but the CTF modulation associated with the contrast transfer, sin g (k) for (e) or cos g (k) for (f) (indicated by broken arrows), induces an irreversible operation. Therefore, to restore the exit wave function of the object from the observed image, a detour has to be taken starting from the two images, (a) and (b) (indicated by solid arrows).

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FIGURE 37. Complex reconstruction results. Small image areas were selected from the original images (from Figure 6 in Danev et al., 2004). Before aligning, the HDC‐TEM image

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is not a usable cross‐correlation peak due to the asymmetric point spread function of HDC‐TEM by the electron‐dose attenuation through the p‐phase plate. The attenuation coeYcient value used was a0 ¼ 0.52. It was calculated as the square of the amount of unscattered electrons for p/2 phase plate at 300 kV, which was 0.72. After demodulation, there is still some directionality in the image. These eVects are unavoidable when using asymmetric phase plates because they are caused by the anisotropy in the transfer of the lower frequency components. The lowest spatial frequencies will be more noticeable perpendicular to the phase plate edge because the edge in this direction is closest to the primary wave focus (zero‐th order beam). By applying the complex reconstruction scheme, the object wave amplitude (Figure 37c) and phase (Figure 37d) were calculated. The amplitude part contains very little information. Only areas with strong scattering displayed amplitude contrast. Most of the object information was contained in the phase part. After reconstruction, the detailed structure of the graphite flake becomes visible. Fine details, which are neither visible in the DPC‐TEM image nor the HDC‐TEM image, were revealed. C. Comparison of Contrast Transfer EYciency Among Various Schemes To make the comparison quantitative among various TEM schemes herein proposed, a novel TEM index, information transfer reliability (ITR), was introduced based on a linear inverse theory originally developed to characterize measurement models (Sugitani et al., 2002). ITRs regarding four observation schemes [two unreplaceable schemes: defocus phase contrast (conventional) and Zernike phase contrast, and two derived schemes: defocus series based on DPC and complex observation] were calculated for the images, respectively, obtained with computer simulation for a protein (see Figure 38a), a high‐potential sulfur protein. Recognition of the 0.5 nm diameter prosthetic group, an iron‐sulfur cluster, included in the protein was used as a criterion for good‐quality images. Based on the argument on the relation between the recognizability by our image perception and the TEM index ITR calculated for the same image, superiority in the observation scheme was quantitatively confirmed for the complex observation and its single experimental version, the ZPC. was demodulated for symmetrization of the CTF by multiplying by i sgn (kx). Then, alignment between two images (a and b0 ) was performed by cross‐correlation. (a) A DPC‐TEM image of graphite particles. (b) An HDC‐TEM image taken from the sample area shown in (a). (b0 ) An HDC‐TEM image demodulated. (c) A reconstructed object wave amplitude. (d) A reconstructed object wave phase.

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1. Linear Forward Theory of Contrast Transfer in TEM As discussed in Section A, we can assume the function form of wave functions at the exit plane as 1 þ a(r) þ ib(r), where 1 corresponds to the incidence of plane wave, a(r) to the absorption, and b(r) to the phase retardance due to the object. When scattered electron waves penetrate through the objective lens, lens‐dependent phase retardance, g(k), arises. The phase plate can add another phase retardance, f, which can be assumed independent of the spatial frequency. Here, all the functions are real valued. The weak object condition reads j1 þ aðrÞj ffi 1;

jbðrÞj 1:

ð71Þ

By manipulating only the scattered electrons with a Zernike phase plate, the image at the signal plane is detected as follows:
 I ðrÞ ¼ j aðrÞ þ ibðrÞ þ eif  FT 1 ½eigðkÞ j2 ffi 1þ þ 2cosffaðrÞ  Re½FT 1 ½eigðkÞ g ð72Þ  2cosffbðrÞ  Im½FT 1 ½eigðkÞ g þ 2sinffaðrÞ  Im½FT 1 ½eigðkÞ g þ 2sinffbðrÞ  Re½FT 1 ½eigðkÞ g The Fourier transform of Eq. (72) is ~I ðkÞ ¼ dðkÞ þ 2cosðgðkÞ  fÞFT ½aðrÞ þ 2isinðgðkÞ  fÞFT ½ibðrÞ:

ð73Þ

Equation (73) suggests that the image in k space is explicitly given in a standard form adequate to the linear theory as d ¼ Gm;

ð74Þ

where d corresponds to the k‐space image data, G to the image transfer matrix defining the observation scheme, and m to wave functions at the exit plane. The form of Eq. (74) allows us to apply the linear inverse theory (Manke, 1989) to the inverse problem in TEM, namely the reproduction of exit‐wave functions. The linearized formalism must finally lead us to a quantitative comparison of diVerent experimental methods. 2. Linear Inverse Theory for TEM Observation The model parameter m, which is the solution of the linear discrete inverse problem formulated with Eq. (74), can be obtained by multiplying the generalized inverse matrix Gg with the obtained data d. Gg looks similar to the usual inverse matrix but is actually diVerent, because Gg is not a square matrix, and, therefore, GgG or GGg does not tend to be a unit matrix. The

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issue of how the generalized inverse matrix is determined and how an otherwise ill‐determined inverse problem may be converted to a well‐ determined one is discussed using a priori information. 3. Generalized Inverse Matrix for TEM Observation We derive the explicit form of the generalized inverse matrix Gg for each of the observation schemes previously discussed. For the complex scheme, the k space expression of the matrix formula derived from Eq. (74) is given as       ~I b ðkÞ  dðkÞ cosgðkÞ isingðkÞ FT½aðrÞ ~I p ðkÞ  dðkÞ ¼ 2 singðkÞ icosgðkÞ FT½ibðrÞ   ð75Þ mr ðkÞ ¼ Gc mi ðkÞ where mr (k) and mi (k) represent the real and imaginary components in the complex image as developed in the previous section. The generalized inverse matrix for the complex scheme is given as   2 cosgðkÞ singðkÞ g : ð76Þ Gc ¼ 4 þ e2 isingðkÞ icosgðkÞ 4. Information Transfer Reliability Derived from Model Resolution Matrix To obtain a matrix formula connecting the estimated (mest) and true (mtrue) model parameters, we have the following formula: d obs ¼ Gmtrue

ð77Þ

mest ¼ G g d obs ¼ G g Gmtrue ¼ Rmtrue

ð78Þ

R ¼ G g G

ð79Þ

and obtained

Here the matrix R is called the model resolution matrix. Because R is exclusively related to the data kernel G and the a priori assumption e, it can be used to evaluate an experimental scheme. In particular, the diagonal elements of the model resolution matrix represent a faithfulness of the estimated model parameters to the true ones. That is to say, when the diagonal elements take values close to 1, the obtained image truly represents the object or the optical information of the object, and the changes in amplitude and phase of incident waves are transferred exactly to the image. Therefore, we can use the diagonal elements of the model resolution matrix

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as an evaluation of the employed observation scheme. We term this new set of parameters ‘‘information transfer reliability’’ (ITR) (Sugitani et al., 2002). For the complex observation scheme, ITR is given as follows: 0 1 0 Rc ðk1 Þ @ A Complex observation : Rc ¼ G g ... c Gc ¼ 0 Rc ðkm Þ ðm : total of digitized frequenciesÞ 4 ITRc ¼ Rc ðkÞ ¼ for the real and imaginary component: 4 þ e2 ð80Þ The damping factor, e2 and, thus, Rc are generally k dependent. For the DPC and the ZPC schemes, they are given as follows: 0 1 0 Rd ðk1 Þ Aðm : total of digitized frequenciesÞ DPC : Rd ¼ @ ... 0 Rd ðkm Þ P 4 sin2 ðgi ðkÞÞ ITRd ¼ Rd ðkÞ ¼ P 2 ð81Þ 4 sin ðgi ðkÞÞ þ e2 0 1 0 Rz ðk1 Þ Aðm : total of digitized frequenciesÞ ZPC : Rz ¼ @ ... 0 Rz ðkm Þ 
P 4 cos2 gi ðkÞ 
ITRz ¼ Rz ðkÞ ¼ P : ð82Þ 4 cos2 gi ðkÞ þ e2 Apart from e2, this formula only includes CTFs (sin gi (k) and cos gi (k)) characteristic of employed observation schemes, which makes ITR a unique measure appropriate to evaluate the quality of the schemes themselves. In reality, however, ITR becomes materially dependent through the a priori factor e2, which can be explicitly defined as the universe of the signal‐to‐noise ratios (SNRs) of target objects. 5. Image Simulations of High‐Potential Sulfur Protein for Four Observation Schemes The TEM image simulations were performed under doses of 1000 e nm2 for the four schemes, conventional (DPC), defocus series based on DPC, ZPC, and complex observation, with a resolution limit of 0.5 nm. The results are shown in Figure 38 (Sugitani et al., 2002).

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FIGURE 38. High‐potential iron sulfur protein images simulated with the four TEM observation schemes (from Figure 4 in Sugitani et al., 2002). The total dose assumed to each scheme was set at 1000 e/nm2 and the higher resolution limit to 0.5 nm. The spherical

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Figure 38 clearly reveals a significant diVerence in image quality depending on the observation scheme. DPC‐TEM (Figure 38b) gave very low contrast and significant image deterioration induced by the sin‐CTF characteristic to this scheme. On the other hand, the ZPC (Figure 38e) produced high contrast, clearly illustrating the overall protein shape, with a maximum contrast at the iron‐sulfur complex. The image shown in Figure 38d is another DPC image simulated with a deeper defocus to enhance the contrast, which is to be combined with the image of Figure 38c (equivalent to Figure 38b) to present the minimum number (two) of a defocus series. The pair of images shown in Figure 38f and g are phase and amplitude images synthesized from the two images shown in Figure 38b and e by complex observation. The SNR of the phase component of the complex image (Figure 38g) is almost identical but slightly worse than that obtained with the ZPC. This is due to the low‐resolution feature used in these simulations, where the cos‐CTF characterizing the image in Figure 38e is suYcient to reproduce the original wave function because of almost‐flat frequency dependence up to the employed resolution limit, which guarantees less image deterioration. The slight reduction in the SNR seen in Figure 38g is due to the combination of the high‐contrast image of Figure 38e with the low‐contrast one of Figure 38b. The complete absence of the signal in Figure 38f, which represents the amplitude component of the complex image, indicates that the test sample is a pure phase object. 6. Wiener Filter-Based TEM Images and Their ITRs Images shown in Figure 38 represent raw data from TEM simulations for four observation schemes. From the viewpoint of SNR enhancement according to the frequency components, raw data must be filtered optimally by taking into consideration the frequency components included. To optimize the overall SNR, the Wiener filter is often employed, which is explicitly given by the form of CTF/(|CTF|2 þ (SNR)1) in our study and defines the observation scheme-dependent ITRs given by Eqs. (80) to (82). Figure 39 shows Wiener‐filtered TEM images, together with ITRs developed in k space and the radial dependence of ITRs for a resolution of 0.5 nm under a dose of 1000 e nm2 (Sugitani et al., 2002). The exit‐wave function

aberration parameter CS was set to 4 mm. (a) A space‐filling model of the high‐potential iron sulfur protein. (b) An image simulated using DPC‐TEM with 100nm underfocus. (c) and (d) Two images simulated using DPC‐TEM with (c) 100 nm and (d) 1000 nm underfocus. (e) An image simulated using ZPC‐TEM with 0‐nm defocus. (f and g) Complementary pair images simulated with DPC‐ and ZPC‐TEM schemes with 0‐nm defocus [both (f) and (g)].

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FIGURE 39. Wiener‐filtered images at a higher resolution limit of 0.5 nm and their ITRs for the four TEM observation schemes (from Figure 7 in Sugitani et al., 2002). First row contains (a) phase shift of the exit‐wave function, (f) power spectrum of the exit‐wave function, and (k) plot of the radial power spectrum. The other rows contain sets of three figures as described below. First column shows images filtered from (b) DPC‐TEM (Figure 38b), (c) DPC‐TEM defocus series (Figure 38c and d), (d) ZPC‐TEM (Figure 38e), and (e) complementary pair imaging (Figure 38f and g). Second column shows ITRs of the corresponding images in k‐space. Third column shows a radial plot of ITRs (corresponding scale is given in the left‐side y‐axis) and CTF‐modulated image power spectra (corresponding scale is given in the right‐side y‐axis), which represents the amount of transferred information.

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(Figure 39a) does show fairly detailed structure, which is also reflected in the Fourier transform (Figure 39f) as a complicated pattern in the widened k space. The general trends described in the previous paragraphs are also evident here; for example, the ZPC scheme (Figure 39d), which gives the highest SNR, also allows clear identification of the iron‐sulfur complex in the bottom center. Together with the lowered ITR, the TEM object index, Info (a real ratio of integrations between two power spectra corresponding to the CTF‐modulated and the exit‐wave function), becomes much smaller for the DPC (Figure 39l) and slightly smaller for the defocus series (Figure 39m). Recognition of the iron‐sulfur complex becomes uncertain in Figure 39b and c. On the other hand, comparison between the wave function (Figure 39a) and the simulated images indicates strongly that the iron‐sulfur complex can be visualized with confidence in the two images simulated with the ZPC (Figure 39d) or complex (Figure 39e) scheme. The systematic background noise, which is the reflection of the fine structure of ITRs in the higher frequency region, also becomes larger in the images shown in Figure 39b and c. Because this noise also overlaps somewhat with the protein image, some discrepancies in the image become clearer between the ZPC (Figure 39d) and the phase image (Figure 39e). Comparing the Fourier‐transformed wave function (Figure 39f ) and the simulated images in k space (Figure 39g to j) reveals a common higher frequency pattern appearing in the four ITRs (Figure 39e to o) that must reflect the k space wave function. This pattern resemblance may arise from the (SNR)1 factor included in the denominator of the ITR formula as e2 in Eqs. (80) to (82).

VIII. DISCUSSION Comparative studies of TEM imaging among diVerent phase contrast schemes developed in this chapter have clearly shown the superiority in contrast to those methods that use phase plates. This result is rather natural from the fact that what can be observed with TEM are phase objects in regard to the electron wave. Therefore, it may not be enough to demonstrate the superiority based on an inspection but better to quantify it from the angle of image analysis. The novel indices introduced in Section VII.C, information transfer reliability (ITR), and info are the eVorts toward the end. In this section, we discuss what is actually improved with phase plate TEM in the term of image quality.

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A. Issue of Electron Loss by the Phase Plate The crucial disadvantage of phase plate TEM is the loss of electrons due to the phase plate, which is clearly illustrated in Figure 9. We have already met trouble brought about by the loss in the restoration of the ZPC image from the HDC, which was mattered as demodulation as shown in Figure 37. The demerit must be much more harmful if the contrast recoverage in the lower frequency components is impaired. To evaluate the electron loss effect, we compared the power spectra of diffractograms between images taken with and without phase plates. A new frequency dependent index, G(k) (Gain), which estimates the power spectral gain yielded by using a phase plate defined as a ratio of corresponding spectral integrations similarly to Info described in §VII‐C, was introduced. The result is shown in Table 6 together with a schematic for the definition (Table 6a). To avoid very low frequency components which are usually uncertain, the integration was performed in the frequency region larger than 0.05 nm1. Due to the definition shown in Table 6a, gain zero means no gain, namely the ratio of two values is one. For the case of negatively stained ferritin images taken with and without a Zernike phase plate, Gain G(k) in the low frequency region is remarkable in every defocus condition employed. Particularly in the near focus condition (the upper trace in Table 6b), the Gain is extraordinary large, say about 10. The Gain becomes smaller when the defocus of the conbentional image becomes deeper. This result has to be expected as the defocus phase contrast becomes higher when it is deeper. For the case of ice‐embedded sample images taken with a Hilbert phase plate, the Gain is not so remarkable (the fourth and fifth row traces in Table 6b). This might be due to the Hilbert phase plate (half plane‐phase plate) twice thicker than the Zernike phase plate, which is naturally leading to the larger loss of electrons. The Hilbert differential method favors the lower frequency to recover as mentioned in §V‐C. A particular frequency that gaves G(k) ¼ 0 (gain zero) is the point where the contrast enhancement due to the phase plate turns to the contrast dehancement due to the electron loss. As already mentioned, the gain profile of G(k), depends on the defocus used for the conventional TEM imaging. Contrast enhancement due to the deep defocus is clearly reflected in the Gain G(k) obtained from the comparison between the near focus Zernike and the deep defocus conventional (see the 3rd row in Table 6b), where the zero crossing point is shifted to the very low frequency region. This deep defocus effect is also evident in the comparison between the Hilbert and the conventional contrast for cyanobacteria (see the fourth row in

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GAIN PROFILE OBTAINED CORRESPONDINGLY TAKEN

TABLE 6 COMPARISON BETWEEN TWO IMAGES PHASE CONTRAST AND CONVENTIONAL TEMS

FROM THE BY

Table 6b), where the zero crossing occurs at the frequency around 0.1 nm1. Keep in mind that an underfocus of 15 m was used for the conventional TEM imaging to maximize the contrast. Nevertheless, the contrast enhancement

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of the Hilbert image is so strikingly percepted compared to the conventional image (see Figure 25), which may indicate the overall view of bacterial images is mostly determined by the frequency component lower than 0.1 nm1 or the geometrical component larger than 10 nm. Contrary to the contrast enhancement, the spatial resolution is dehanced by phase plates. Possible causes for the dehancement are: i) the electron loss as above discussed; ii) charging effect exaggerated in the higher frequency region; and iii) the lowered optimum cut‐off for the cosine‐CTF, as discussed in §IV‐A. Overall, we can recognize that our 300 kV phase contrast TEM enhance the information particularly in the frequency range smaller than 0.3 nm1 with an aid of phase plates.

B. Issue of Weak Objects The requirement of the weak object condition has often been mentioned for phase contrast schemes to work as theory predicts. Although thick samples may tend to belong to strong objects, the condition is not as obvious as we might think. Let us think, for example, the case of a uniformly thick ice. Most incident electrons may be inelastically scattered by the ice object with single or multiple collisions, but the part of elastic scattering, even though small fractions, could contribute to the phase contrast. To the specimen embedded in ice, this situation is equivalent to having an electron irradiation with lessened intensity and increased background noise. Particularly when the zero‐loss image is taken, the uniformly thick ice behaves simply as an incidence attenuator. In such a case, the weak object condition can be applied to fairly thick samples. Another clue to ease the weak object condition for thick samples is the averaged thickness. In a situation that only a small portion of the observed area is occupied by very thick samples leaving the remaining area unoccupied, the 0‐th order beam acting as an internal reference is still overwhelming if the beam coherency is high enough, which guarantees adequate phase contrast. Both cases are implied by the condition claimed in Eq. (45) but have never been explicitly discussed. Unexpectedly, successful results for very thick samples such as whole cells or whole bacteria shown in section IV or V with ZPC‐TEM or HDC‐TEM must be interpreted along this argument. In other words, even for objects usually classified as strong, the coherent low‐frequency components guaranteeing the phase contrast must be large enough to match with the other

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components contributing to the background, once inelastically scattered electrons are carefully removed with an energy filter.

C. Issue of Specimen Charging Although it is not as crucial as experienced in the case of the phase plate charging, the specimen charging also matters in a specific manner in the phase contrast scheme using phase plates. Charging of specimens is generally problematic in the biological study with electron crystallography. In the biological TEM, which does not claim the spatial resolution so high, however, it is usually disregarded because the primary eVect of charging, when it is not quite non‐uniform in the specimen, is a deflection of electrons. This kind of eVect does appear implicitly as an overall spatial shift of images and then can be easily overlooked. In the case of phase plate TEM, the deflection kind of eVect is quite serious because it misaligns the direct beam (primary wait) from the proper setting, for example, from the alignment to the center hole of the Zernike phase plate. Actually, the specimen charging was first severely recognized in the HDC‐TEM experiment for the resin‐embedded sectioned cell shown in Figure 19. Thin specimens such as negatively stained ferritins shown in Figure 13 did not show any such a drastic eVect. These results could be explained by the great diVerence in the extent of specimen charging between the two samples. The misalignment of the direct incidence induced by the specimen charging could be compensated by the realignment of the beam, but the procedure is occasionally tedious. The same remedy employed to phase plate charging could be applied to specimen charging. For the resin‐embedded samples, the carbon coating to the open side opposite to the side facing to the supporting film is already known to be useful and widely employed. In our studies with ice‐embedded samples, sandwiching specimens with two carbon films supported with respective two TEM grids has been found to be very eVective to kill the charging. The procedure is rather straightforward, as only one step was added to the conventional procedure of rapid freezing of an aqueous suspension as the second carbon film holding TEM grid was attached and pressed to the surface of the suspension. Of course, this procedure is not always called for if researchers are laborious enough to take the eVort of the realignment task. In the application of the phase plate TEM to tomography, however, the specimen preparation with the sandwiching method might be important.

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IX. CONCLUSIONS A typical reaction, I have met so far when I have shown high‐contrast images taken with phase plate TEM to people, was a surprise first and a suspicion next. The surprise could be natural, but the suspicion may vary depending on one’s experience in TEM. People belonging to end‐users of TEM tended to consider that even conventional TEM could do the same thing if such a crude idea as inserting one simple phase plate into the TEM column could work so drastically on the contrast. Experienced TEM users rather doubted the phase plate insertion itself, as they considered it so harmful for getting clean images. The former suspicion could be fairly dispelled by this review. The latter one, however, is still diYcult to be cleared because from the quantitative viewpoint, there are evils in phase plate TEM, such as the loss of electrons and the lowered cutoV frequency, coupled with remaining minute charges. To dispel this suspicion, therefore, further development of phase plate technology is mandatory. For example, the phase retardance due to the matter (carbon) could be replaced by phase retardance due to the non‐matter such as electric or magnetic field. But this belongs to another story, and we are able to conclude as follows: 1. Phase contrast schemes with phase plates such as the Zernike phase plate and the half‐plane p‐phase plate work as theory predicts. 2. TEM images taken with an acceleration voltage larger than 100 kV can be quite enhanced in the image contrast with use of phase plates. 3. Particularly, 300‐kV HDC‐TEM images have a great advantage to provide high‐contrast images for the unstained ice‐embedded biological samples. 4. The long‐standing issue of the phase plate charging can be settled with the carbon coating conducted to the whole phase plate at the very last stage of the fabrication procedure. 5. Two issues of the loss of electrons due to phase plates and the remaining charge eVect, particularly revealing in the high frequency end, have to be settled in the near future.

ACKNOWLEDGMENTS I owe the development and biological applications of phase contrast TEM with phase plates to the following collaborators:

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Development: Radostin Danev, Rasmus Schroeder, Shozo Sugitani, Hiroshi Okawara, Toshiyuki Itoh, Toshikazu Honda, Toshiaki Suzuki, Yoshiyasu Harada, Yoshihiro Arai, Fumio Hosokawa, Sohei Motoki, and Kazuo Ishizuka Applications: Nobutaru Usuda, Ayami Nakazawa, Kiyokazu Kametani, Masashi Tanaka, Hideo Hirokawa, Fumio Arisaka, Koki Taniguchi, Holland Cheng, Xing Li, and Kenneth Holmes This work was supported in part by a Grant‐in‐Aid for Creative Scientific Research (No. 13GS0016) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. REFERENCES Angert, I., Burmester, C., Dinges, C., Rose, H., and Schroeder, R. (1996). Elastic and inelastic scattering cross‐sections of amorphous layers of carbon and vetrified ice. Ultramicroscopy 63, 181–192. Badde, H. G., and Reimer, L. (1970). Der Einflub einer streuenden Phasenplatte auf das elektronen mikroskopische Bild. Z. Naturforschg. 25a, 760–765. Balossier, G., and Bonnet, N. (1981). Use of electrostatic phase plate in TEM. Transmission electron microscopy: Improvement of phase and topographical contrast. Optik 58, 361–376. ¨ ber die Kontraste von Atomen in Electronenmikroskop. Z. Naturforschg. Boersch, H. (1947). U 2a, 615–633. Cheng, H., Danev, R., Xiang, L., and Nagayama, K. (2004). Unpublished data. Danev, R., and Nagayama, K. (2001a). Complex observation in electron microscopy. II. Direct visualization of phases and amplitudes of exit wave functions. J. Phys. Soc. Jpn. 70, 696–702. Danev, R., and Nagyama, K. (2001b). Transmission electron microscopy with Zernike phase plate. Ultramicroscopy 88, 243–252. Danev, R., and Nagayama, K. (2004). Complex observation in electron microscopy. Reconstruction of complex object wave from conventional and half plane phase plate image pair. J. Phys. Soc. Jpn. 73, 2718–2724. Danev, R., Okawara, H., Usuda, N., Kametani, K., and Nagayama, K. (2002). A novel phase‐ contrast transmission electron microscopy producing high‐contrast topographic images of weak objects. J. Biol. Phys. 28, 627–635. Danov, K., Danev, R., and Nagayama, K. (2001). Electric charging of thin films measured using the contrast transfer function. Ultramicroscopy 87, 45–54. Danov, K., Danev, R., and Nagayama, K. (2002). Reconstruction of the electric charge density in thin films from the contrast transfer function measurements. Ultramicroscopy 90, 85–95. Faget, J., Fagot, M., Ferre, J., Fert, C. (1962). Microscopie Electronique a Contraste de Phase. Proceedings of the 5th International Congress Electron Microscopy A‐7. New York: Academic Press. Fogg, G. E., Stewart, W. D. P., Fay, P., and Walsby, A. E. (1973). The Blue‐Green Algae. London: Academic Press. Fernandez‐Moran, H. (1960). Low‐temperature preparation techniques for electron microscopy of biological specimens based on rapid freezing with liquid helium II. Ann. NY Acad. Sci. 85, 689–713.

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